Prediction of Limit Cycle Oscillation in an Aeroelastic System … · 2020. 1. 18. · Prediction of Limit Cycle Oscillation in an Aeroelastic System using Nonlinear Normal Modes

Prediction of Limit Cycle Oscillation

in an Aeroelastic System using Nonlinear Normal Modes

Christopher Wyatt Emory

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Aerospace Engineering

Mayuresh J. Patil, Chair

Robert A. Canfield

Charles M. Denegri, Jr.

Rakesh K. Kapania

December 3, 2010

Blacksburg, Virginia

Keywords: Nonlinear Aeroelasticiy, Limit Cycle Oscillation, Nonlinear Normal Modes

Copyright 2010, Christopher Wyatt Emory

Prediction of Limit Cycle Oscillation

in an Aeroelastic System using Nonlinear Normal Modes

Christopher Wyatt Emory

(ABSTRACT)

There is a need for a nonlinear flutter analysis method capable of predicting limit cycle

oscillation in aeroelastic systems. A review is conducted of analysis methods and experi-

ments that have attempted to better understand and model limit cycle oscillation (LCO).

The recently developed method of nonlinear normal modes (NNM) is investigated for LCO

calculation.

Nonlinear normal modes were used to analyze a spring-mass-damper system with

nonlinear damping and stiffness to demonstrate the ability and limitations of the method

to identify limit cycle oscillation. The nonlinear normal modes method was then applied to

an aeroelastic model of a pitch-plunge airfoil with nonlinear pitch stiffness and quasi-steady

aerodynamics. The asymptotic coefficient solution method successfully captured LCO at

a low relative velocity. LCO was also successfully modeled for the same airfoil with an

unsteady aerodynamics model with the use of a first order formulation of NNM. A linear

beam model of the Goland wing with a nonlinear aerodynamic model was also studied. LCO

was successfully modeled using various numbers of assumed modes for the beam. The concept

of modal truncation was shown to extend to NNM. The modal coefficients were shown to

identify the importance of each mode to the solution and give insight into the physical nature

of the motion.

The quasi-steady airfoil model was used to conduct a study on the effect of the

nonlinear normal mode’s master coordinate. The pitch degree of freedom, plunge degree of

freedom, both linear structural mode shapes with apparent mass, and the linear flutter mode

were all used as master coordinates. The master coordinates were found to have a significant

influence on the accuracy of the solution and the linear flutter mode was identified as the

preferred option.

Galerkin and collocation coefficient solution methods were used to improve the results

of the asymptotic solution method. The Galerkin method reduced the error of the solution

if the correct region of integration was selected, but had very high computational cost. The

collocation method improved the accuracy of the solution significantly. The computational

time was low and a simple convergent iteration method was found. Thus, the collocation

method was found to be the preferred method of solving for the modal coefficients.

iii

Contents

1 Introduction 1

2 Literature Review 3

2.1 SEEK Eagle LCO Flight Testing Experiences . . . . . . . . . . . . . . . . . 3

2.2 LCO Analysis Methods and Experiments . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Pitch-Plunge Airfoil with Structural Nonlinearities . . . . . . . . . . 5

2.2.2 Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Nonlinear Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Nonlinear Normal Modes 18

3.1 Second-Order Formulation

with Asymptotic Modal Coefficient Solution . . . . . . . . . . . . . . . . . . 18

3.1.1 Linear Spring-Mass-Damper Example . . . . . . . . . . . . . . . . . . 22

3.1.2 Alternative Eigenvalue Solution to Linear Coefficients . . . . . . . . . 27

3.1.3 Linear Normal Mode (LNM) Solution . . . . . . . . . . . . . . . . . . 30

3.2 First-Order Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Galerkin Modal Coefficient Solution . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Collocation Modal Coefficient Solution . . . . . . . . . . . . . . . . . . . . . 35

4 Physical Models 37

4.1 Two Degree-of-Freedom Van der Pol Style Oscillator . . . . . . . . . . . . . 38

4.2 Nonlinear Pitch-Plunge Airfoil with Quasi-Steady Aerodynamics . . . . . . . 38

iv

4.3 Nonlinear Pitch-Plunge Airfoil with Unsteady Aerodynamics . . . . . . . . . 41

4.4 Goland Beam Wing with Nonlinear, Quasi-Steady Aerodynamics . . . . . . . 43

5 Results 47

5.1 Two Degree-of-Freedom

Van der Pol Style Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Pitch-Plunge Airfoil

with Quasi-steady Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.1 Sample Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.2 Master Coordinate Study . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.3 Galerkin Modal Coefficient Solution . . . . . . . . . . . . . . . . . . . 74

5.2.4 Collocation Modal Coefficient Solution . . . . . . . . . . . . . . . . . 81


with Unsteady Aerodynamics Sample Results . . . . . . . . . . . . . . . . . 87

5.4 Goland Beam Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4.1 Modal Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Conclusions 105

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Bibliography 109

v

List of Figures

3.1 Three Degree-of-Freedom Spring-Mass-Damper System . . . . . . . . . . . . 23

4.1 Nonlinear Spring-Mass-Damper . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Pitch-Plunge Airfoil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Goland Wing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1 Nonlinear SMD Phase Space Diagrams for First NNM, Case 1 . . . . . . . . 52

5.2 Nonlinear SMD Phase Space Diagrams for Second NNM, Case 1 . . . . . . . 53

5.3 Nonlinear SMD Phase Space Diagrams for First NNM, Case 2 . . . . . . . . 54

5.4 Nonlinear SMD Phase Space Diagrams for Second NNM, Case 2 . . . . . . . 55

5.5 Nonlinear SMD Time History for First NNM, Case 1 . . . . . . . . . . . . . 56

5.6 Nonlinear SMD Time History for Second NNM, Case 1 . . . . . . . . . . . . 57

5.7 Nonlinear SMD Time History for First NNM, Case 2 . . . . . . . . . . . . . 58

5.8 Nonlinear SMD Time History for Second NNM, Case 2 . . . . . . . . . . . . 59

5.9 LCO Time History Plots of Quasi-steady Airfoil at 1.05 Vf . . . . . . . . . . 63

5.10 LCO Phase Space Plots of Quasi-steady Airfoil at 1.05 Vf . . . . . . . . . . 64

5.11 Damped Mode Time History Plots of Quasi-steady Airfoil at 1.05 Vf . . . . 65

5.12 Growth of LCO Motion in Exact System due to Approximate Initial Condition 66

5.13 LCO Phase Space Plots of Quasi-steady Airfoil at 1.17 Vf . . . . . . . . . . 67

5.14 Frequency vs. Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.15 Amplitude vs. Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.16 Asymptotic Convergece at 1.01Vf . . . . . . . . . . . . . . . . . . . . . . . . 69

vi

5.17 Asymptotic Convergece at 1.05Vf . . . . . . . . . . . . . . . . . . . . . . . . 69

5.18 Master Coordinate Amplitude Error Results . . . . . . . . . . . . . . . . . . 73

5.19 Master Coordinate Frequency Error Results . . . . . . . . . . . . . . . . . . 74

5.20 Galerkin with Integration Region 0.5×Exact Amplitudes, 1.17Vf . . . . . . 76





5.25 Galerkin with Integration Region 1.2×Exact Amplitudes, 1.5Vf . . . . . . . 82

5.26 Collocation Modal Coefficient Solution Sample Results at 1.17Vf . . . . . . . 84

5.27 Motion Growth with Collocation Solution . . . . . . . . . . . . . . . . . . . 86

5.28 Collocation Iteration 1 at 1.5Vf . . . . . . . . . . . . . . . . . . . . . . . . . 89



5.31 Steady LCO results at u = 1.05Vf for Linear Flutter Mode - US Aero . . . . 93

5.32 Steady LCO results at u = 1.17Vf for Linear Flutter Mode - US Aero . . . . 94

5.33 Normalized Unsteady (US) vs. Quasi-steady (QS) LCO solution at 1.17Vf . 95

5.34 Goland Wing with 5 Bending and 5 Torsion Modes . . . . . . . . . . . . . . 97

5.35 Goland Wing with Multiple Modal Solutions . . . . . . . . . . . . . . . . . . 101

5.36 Goland Wing LCO Amplitude Error vs. Number of Modes . . . . . . . . . . 102

vii

List of Tables

5.1 NNM Results for SMD with Nonlinear Damping . . . . . . . . . . . . . . . . 51

5.2 Galerkin Integration Region Data . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Collocation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 Collocation Iteration at a High Velocity . . . . . . . . . . . . . . . . . . . . . 88

5.5 Beam Wing Flutter Velocities and Run Speeds . . . . . . . . . . . . . . . . . 100

5.6 Beam Wing Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.7 Beam Wing CPU Times in Seconds . . . . . . . . . . . . . . . . . . . . . . . 101

5.8 Beam Wing Modal Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.9 Normalized Beam Wing Modal Coefficients . . . . . . . . . . . . . . . . . . . 104

viii

Chapter 1

Introduction

Aeroelasticity is the interaction of aerodynamic, elastic, and inertial forces. Flutter is an

aeroelastic phenomenon in which a dynamic instability occurs due to the interaction of

aircraft surfaces, most often wings, with the air flowing past. In classical flutter, the wing

will cyclically increase in amplitude until it breaks which can occur in very few cycles. A

number of modern aircraft, such as the F-16, have also experienced a phenomenon termed a

limit cycle oscillation (LCO) in which the wing behaves similarly to a flutter situation, but

does not diverge. One or more nonlinearities, such as geometric, aerodynamic, stiffness, or

structural damping, in the system act to limit the amplitude of the motion. In the case of the

F-16, the occurrence of LCO can make it difficult or impossible for the aircrew to perform

necessary tasks. If the amplitude is large enough, the motion can damage the aircraft or

stores attached to the wings[1].

Flutter has typically been modeled with linear analysis. Since LCO is by its very

nature nonlinear, these linear models are not capable of predicting all aspects LCO. Linear

flutter analysis has been demonstrated to be adequate to predict the frequency and modal

composition, but it is incapable of identifying the onset velocity or the amplitude[2]. This

inability to fully predict the LCO creates the need for extensive flight testing. For an aircraft

like the F-16, the number of under-wing store configurations has been increasing and flight

testing costs are continually rising[3]. Denegri identifies the need for a nonlinear flutter

1

Chapter 1. Introduction

analysis method capable of predicting all the aspects of LCO behavior[2][3].

Many models and experiments, covered in detail in the following literature review,

have been pursued to aid in the development of a model capable of capturing the physics

involved in LCO. Many of these models are purely computational. Most simply are not

practical for modeling a real world case like the F-16 due to the large amount of computation

necessary. One of the most promising modeling approaches has been harmonic balance and

it has been applied to a few F-16 test cases with somewhat favorable results[4].

The goal of this work is to take a step toward a new, practical method of nonlinear

analysis to model LCO that offers insight into the physics underlying the LCO phenomena.

Furthermore, the analysis methodology should be extendable to complex real world problems

in the future. The method of nonlinear normal modes (NNM), which was developed during

the 1990’s, is explored as an alternative method of LCO prediction. The NNM model can

reduce the order of the system to a single degree of freedom while trying to maintain some or

all nonlinearity and the effect of all degrees of freedom desired in the model. Only having to

solve a system with a single degree of freedom could save considerable computational time.

The modal analysis nature of NNM should also offer some unique information about the

physical modeshapes involved in LCO motion.

Terminology used in the description of flutter and LCO is often ambiguous and con-

fusing. Many papers use the same terms to mean different things. As such, in this paper

the following definitions will be used throughout unless explicitly noted. LCO will refer to a

sustained periodic oscillation due to nonlinear aeroelastic interaction. Linear flutter or clas-

sic flutter refers to divergent aeroelastic oscillations in a linear system or model. Nonlinear

flutter refers to divergent aeroelastic oscillation in a nonlinear system or model. A flutter or

LCO boundary refers to a line or curve that dictates the onset of the particular aeroelastic

phenomenon in a freestream velocity and other variable space.

2

Chapter 2

Literature Review

This literature review is composed of three sections. The first section reviews papers from the

US Air Force SEEK Eagle office about their experiences with limit cycle oscillation (LCO) in

Flight Testing of the F-16. The following section reviews a number of modeling techniques

and experiments that other researchers have used to study flutter and LCO. The final section

briefly reviews the development and implementation of nonlinear normal modes.

2.1 SEEK Eagle LCO Flight Testing Experiences

Bunton and Denegri[1] discuss the general characteristics of LCO encountered in the F-16

and the basic problems that are caused. LCO is found to occur at high subsonic and transonic

speeds. The response is characterized by antisymmetric motion of the wings. The behavior

has been observed in both straight-and-level flight and elevated load factor maneuvers. The

motion is self sustaining and displays hysteresis with respect to the onset airspeed and load

factor. Typically, LCO has a fixed amplitude for a given set of flight conditions and the

amplitude increases with airspeed. A number of different LCO behaviors have been observed

as the load factor is increased. In some cases, the amplitude increases gradually as the load

factor is increased while in others the amplitude suddenly drops off at a higher load factor. It

is also possible for the LCO to suddenly appear at a high load factor and then grow rapidly

3

Chapter 2. Literature Review

with further increased load factor. The LCO can make it difficult or impossible for the

aircrew to perform necessary tasks and if the LCO amplitude is large enough, it can damage

the aircraft or stores carried under-wing.

Denegri[2] demonstrates that linear flutter analysis is adequate to predict LCO fre-

quency and modal composition, but is unable to predict the onset velocity or amplitude of

LCO. Typical LCO is defined by a gradual onset of sustained oscillation that has a con-

stant amplitude for given flight conditions. The amplitude of typical LCO increases with

an increased airspeed. Nontypical LCO is defined as sustained oscillations that are only

present over a limited speed range, first appearing and then disappearing when airspeed is

increased. Typical LCO, nontypical LCO, and classic flutter are all shown to occur in F-16

flight tests. It is hypothesized that the antisymmetry of the LCO motion is due to the ease

of energy transfer in antisymmetric fuselage body modes as opposed to the high resistance

in symmetric fuselage body modes. Denegri[2] conveys the need for a nonlinear flutter anal-

ysis method capable of discerning the difference between typical LCO, nontypical LCO, and

classical flutter behaviors discussed.

Denegri et al.[3] studied the progression of the LCO mode shape in the F-16 wing as

flight speed and load factor were increased. Flight tests of the F-16 show that at the onset

of LCO the mode shape bears a strong resemblance to the linear flutter mode shape from

analysis. The mode shape becomes increasingly nonsynchronous (more galloping character)

as the speed, and therefore the amplitude, was increased. At a high speed and load factor,

the motion was seen to become nearly synchronous while maintaining LCO which indicates a

change in mechanism. The authors hope their observations can help shed light on the physical

mechanisms involved in the nonlinear behavior. They further state that it is necessary to

find methods to accurately predict flutter and LCO characteristics because the cost of flight

testing is growing and the number and combinations of store configurations are continually

increasing requiring more flight tests to insure the safety of pilots and equipment.

Dawson and Maxwell[5] present a study of asymmetric store configurations on the

F-16. It is shown that asymmetric store configurations can lead to a higher or lower critical

4


flutter and LCO speed than symmetric installations of each half of the asymmetric configu-

ration. This proves that the industry practice of using half span models and assuming the

more critical case can unnecessarily limit the flight envelope or be catastrophically dangerous

if a low flutter speed is not predicted. Flight tests demonstrate that the LCO behavior can

be better or worse for an asymmetric configuration than the symmetric halves. It makes

obvious the need for models that are able to capture the complex behavior in an asymmetric

aircraft configuration.

2.2 LCO Analysis Methods and Experiments

2.2.1 Pitch-Plunge Airfoil with Structural Nonlinearities

The aeroelastic behavior of a two degree of freedom, pitch-plunge airfoil with a nonlinear

stiffness in one or both degrees of freedoms has been the focus of many analyses. The first

nonlinear investigation of this model was conducted by Woolston et al.[6] through the use

of an analog computer. They studied nonlinearities due to freeplay, hysteresis, a soft cubic

pitch spring, and a hard cubic pitch spring. For the hard cubic spring, the LCO boundary

did not change as compared to the linear flutter boundary and LCO was predicted beyond

the linear flutter velocity. The LCO amplitude increased as the velocity was increased. For

the soft cubic spring, the nonlinear flutter motion was highly divergent beyond the linear

flutter velocity. Furthermore, the nonlinear flutter onset velocity decreased as the initial

condition amplitude was increased. Beyond a small initial condition, the freeplay nonlinearity

significantly decreased the nonlinear flutter speed. Over a moderate initial condition range,

the system experienced LCO and beyond the moderate range divergent nonlinear flutter

was encountered. Experimental results were obtained for the freeplay case and compared

well with the analog computer simulation. For the case with hysteresis, the linear flutter

boundary was again reduced for large enough initial conditions forming a different nonlinear

flutter boundary. For speeds between the lower nonlinear onset velocity and the linear

flutter velocity, the system experienced LCO. Divergent flutter was experienced beyond the

5


linear flutter boundary. Overall, the nonlinear flutter speed did not change from the linear

flutter speed for small disturbances. For larger initial conditions, the nonlinear flutter speed

decreased for all cases except the hard cubic spring.

Lee and LeBlanc[7] studied the effects the airfoil/air mass ratio, undamped plunge

and pitch natural frequencies, distance between the center of mass and the elastic axis,

and multiple cubically nonlinear pitch stiffnesses. Their analysis was accomplished via time

marching of the nonlinear differential equations with Houbolt’s[8] implicit finite difference

method. The time marching scheme used in this paper was later used in many others.

An airfoil with bilinear and cubic nonlinearities in pitch was investigated by Prince

et al.[9]. They used Wagner’s function for the aerodynamics and the nonlinear differential

equations were integrated with the finite difference method of Houbolt[8]. The equations

were also solved with the harmonic balance method which was originally derived by Krylov

and Bogoliubov[10]. The effect of a moment preload on the airfoil was studied. The results

showed LCO well below the linear flutter velocity for both bilinear and cubically nonlinear

stiffnesses. The finite difference method identified a secondary bifurcation, which was not

found by the harmonic balance method due to its assumption that the fundamental harmonic

dominates the behavior of the system. Chaotic motion was shown in particular cases for the

cubic system showing that chaos can occur in the continuously nonlinear system.

The chaotic motion of the airfoil with a cubically nonlinear stiffness was also studied

by Zhao and Yang[11]. The harmonic balance method and fourth order Runge-Kutta was

used to search for chaotic motion. Chaotic motion was found only for a limited range of elastic

axis positions and only when the freestream velocity was in excess of the static divergence

speed.

Lee, Gong, and Wong[12] introduced four new aerodynamic variables to eliminate

the aerodynamic lag integrals in the aeroelastic equations of motions. Earlier work had

solved the Wagner function integrals at each time step for the unsteady aerodynamics. With

the elimination of the integrals the system could be written in terms of eight first-order

ordinary differential equations in the time domain making analytical analysis of the equations

6


possible. For this study, only harmonic solutions to these equations were considered. The

method of slowly varying amplitude was used to investigate the behavior of the system.

Equilibrium points were computed for the harmonic solution and a linear stability analysis

was performed to determine stability of the equilibrium points. The authors provided a

technique to determine the amplitude-frequency relationship. Their results were obtained

by assuming the effects of high harmonics were small and amplitudes were slowly varying

functions of time. The results compared well with fourth-order Runge-Kutta and the Lee

and LeBlance[7] numerical time integration of exact equations.

The eight first-order differential equation form was further used by Lee, Jiang, and

Wong[13] to study both soft and hard cubic springs. For a soft spring, the nonlinear flutter

speed decreases with increases in the pitch and plunge initial conditions. For a hard spring,

the nonlinear flutter boundary is independent of the initial conditions and beyond the flutter

boundary limit cycle oscillation occurs instead of divergent flutter. LCO amplitude increases

as the speed is increased. For a hard spring, an approximate method was used to estimate

the frequency and agreement with Runge-Kutta simulation was found to be dependent on

the plunge-pitch frequency ratio. An asymptotic theory that was a function of the estimated

frequency was used to find the LCO amplitude. Results were good for small frequency but

deteriorated for larger frequency values. The authors suggested that the results could be

improved with the application of center manifold theory.

Center manifold theory was applied to the nonlinear airfoil by Liu, Wong, and Lee[14].

The pitch-plunge airfoil equations were formulated in the eight first-order differential equation

form. The center manifold theory was then applied to reduce the order of the system from

eight to two states. The reduced system was then studied with varying linear and nonlinear

stiffness coefficients. The results of the reduced system closely matched the behavior of a

simulation of the original equations. The reduced order results compared well near the linear

flutter velocity where the center manifold was formulated. However, the results deteriorated

as velocity was increased above the linear flutter velocity. The general behavior of reduced

system’s LCO was comparable to that observed in other studies.

7


With the eight first-order equation form, Liu and Dowell[15] studied the effect of

adding higher harmonics to the harmonic balance method. A secondary bifurcation was

detected via time stepping when the flow velocity was increased past the initial bifurcation at

the linear flutter velocity. The velocity at which the secondary bifurcation occurred depended

on the initial condition. The authors show that secondary bifurcation cannot be identified

with a quasi-steady aerodynamic assumption. The secondary bifurcation was thoroughly

examined with Runge-Kutta integration to create region of attraction diagrams over a range

of initial conditions. The effect of changing the number of harmonics included in the harmonic

balance method was then examined. The results indicate that at least nine harmonics must

be included to detect the secondary bifurcation. As the harmonics were increased beyond

nine, the results better matched the numerical simulation. The results with a higher number

of harmonics give a more precise speed of the secondary bifurcation. The harmonic balance

method only identifies the multiple steady motions and cannot determine the effect of initial

conditions or the stability of the steady motions identified. All branches identified with time

stepping are considered stable. A perturbation analysis was used to determine the stability of

the branches identified in the harmonic balance method. Perturbation of a linearized system

was shown to be adequate to identify the stable motions.

In a follow up study, the high dimensional harmonic balance method (HDHB) was used

to study the secondary bifurcation[16]. HDHB is a reformulation of the harmonic balance

method in terms of time domain variables. The dependent variables are motion values at a

number of equally spaced sub-time intervals over one cycle. This circumvents the necessity

of having to derive the analytical expression of the Fourier coefficients. The HDHB variables

are related to the harmonic balance variables by a constant Fourier transformation matrix.

At the secondary bifurcation the amplitude almost doubled, and the pitch motions switched

from one peak per cycle to three, while the plunge remained at one peak per cycle. The results

of the HDHB compared well with Runge-Kutta results. For small amplitude motion near

the linear flutter velocity, the harmonic balance and the HDHB were close to time marching

results. Away from this point the difference between the harmonic balance, HDHB, and

8


Runge-Kutta results became significant. For an equal number of harmonics included, the

harmonic balance is more accurate than the HDHB. To get the same order accuracy, the

HDHB must include approximately twice as many harmonics. However, harmonic balance

is difficult to derive with a large number of harmonics and HDHB is easy to derive and

implement even with a very large number of harmonics included.

Nonlinear Aeroelastic Test Apparatus (NATA)

The NATA model has been used for many nonlinear studies of the classic two-dimensional

pitch-plunge symmetric airfoil model. A unique test carriage was developed for the low

speed wind tunnel at Texas A&M. The carriage allows for variation of the nonlinear pitch

and plunge stiffness through the use of custom shaped cams. The carriage has to plunge

with the wing, but only the wing pitches creating the only major difference from the classic

two degree of freedom model and experiments. This is easily dealt with by a simple addition

to the mass term in the standard plunge equation.

For one of the early experiments with the NATA model[17], the equations of motion

included a Coulomb damping term and retained nonlinearities in structural stiffness and

kinematics. The aerodynamics were modeled using an approximation of Wagner’s function

which includes a convolution integral. The simulation of the equations was accomplished

with the finite difference scheme of Houbolt[8] using a recurrence formula for the convolution

integral. Experiments found stall flutter of the model with linear stiffnesses to occur at a

pitch amplitude of 0.25 radians. The nonlinear stiffness resulted in LCO pitch amplitudes

sufficiently below the stall flutter amplitude; therefore, the aerodynamics are assumed to be

linear. The LCO is thus assumed to be caused by structural nonlinearity alone. The stability

boundary was shown to be sensitive to initial conditions if Coulomb damping was included

and was not affected if only viscous damping was modeled. The LCO was shown to have the

same amplitude and frequency for a given freestream velocity regardless of initial conditions.

The amplitude of the LCO was shown to increase with an increase in freestream velocity.

Freestream velocity had no measurable effect on the experimental LCO frequency, but the

9


predicted frequency showed a gradual increase. The prediction of LCO amplitude showed

the proper trends, but the magnitudes differed by an appreciable amount.

Gilliatt et al.[18] modeled the NATA system with cubically nonlinear, quasi-steady

aerodynamics and cubically nonlinear spring stiffness. A fourth-order Runge-Kutta method

with a variable time step was used to simulate equations. The initial conditions used were

always displacements and not velocities. Experiments with a linear structure indicated that

stall flutter occured at a pitch amplitude of 0.25 radians. LCO measured for the nonlinear

stiffness case had amplitudes well below the stall flutter amplitude. LCO was observed over a

ten meter per second airspeed band with little change in amplitude. The results agreed with

experiments and analysis performed previously. LCO is said to be due to nonlinear structure

because pitch amplitude is within the area where linear aerodynamics are applicable. Sensi-

tivity of the flutter/LCO boundary to the initial conditions was observed and attributed to

coulomb damping. The resulting LCO amplitude was not a function of the initial conditions.

A fast Fourier transform analysis of the data showed higher harmonics were present in the

motions, especially in plunge degree of freedom. The predicted fundamental frequency was

a little higher than measured. The authors attempted to study internal resonance in the

system but conveyed the difficulty of understanding the phenomenon in terms of the original

dynamic equations.

Gilliatt et al.[19] demonstrated that internal resonances are theoretically possible in

the NATA model with a linear stiffness and quasi-steady aerodynamics with a cubic stall

model.

O’Neil and Strganac[20] modeled the NATA apparatus with viscous and Coloumb

damping, kinematic nonlinearities, centripetal acceleration, and transcendental terms all

included in the equations of motion. For aerodynamics, Theodorsen’s equations were used

with C[k] = 1 leading to quasi-steady aerodynamics. Both pitch and plunge stiffnesses were

nonlinear, but the pitch nonlinearity was much larger as compared to the plunge nonlinearity.

The equations of motions were integrated following the approach used by Lee and LeBlanc[7]

where the finite difference approximation of Houbolt[8] was used. The reduced frequency

10


was observed to be small enough that quasi-steady aerodynamic assumption was valid (k

approximately 0.1). Experiments were done with linear and nonlinear stiffness. The tests

were conducted by holding the model at an initial condition and releasing it in a stabilized

freestream velocity. Experiments were conducted to validate the zero freestream velocity

prediction and at a freestream velocity slightly greater than linear flutter velocity. Slight

differences attributed to small unmodeled nonlinearities were noted between predicted and

actual response for the linear case. Results showed that flutter was suppressed as the elastic

axis was moved toward the center of gravity, effectively mass balancing the system. Nonlinear

spring coefficients were found by a cubic curve fit. Prediction at a freestream velocity slightly

greater than linear flutter velocity showed LCO, but pitch amplitude was too large to compare

to experiment. A higher nonlinear pitch stiffness coefficient was used to keep the amplitude

in linear aerodynamic bounds. Higher freestream velocity was observed to increase LCO

amplitude. LCO onset showed sensitivity to initial conditions which was credited to presence

of Coulomb damping. Analytic results showed good agreement with the experiment and are

consistent with findings of other studies.

Thompson and Strganac[21] added an under wing store to the NATA model. Quasi-

steady aerodynamics were used with equations allowing for nonlinear stiffness, viscous damp-

ing, Coulomb damping, nonlinear kinematic terms. The analysis used a linear plunge stiffness

and a fifth order polynomial for pitch stiffness. Sine and cosine in the nonlinear kinematic

terms were replaced with Taylor series expansions to the third order. This study used the

method of variation of parameters to analyze the system and show the advantage of including

kinematic nonlinearities. It was quickly seen that the kinematic nonlinearities are important

to the response of the system. The equations of motion were numerically integrated ignoring

structural damping and with a small freestream velocity in an attempt to study internal

resonance in the system. The pitch stiffness was held constant while the plunge stiffness

varied to provide varied frequency responses. A slow modulation was observed when a 1:3

frequency ratio was prescribed which is indicative of internal resonance. It is still unclear if

there is a link between store-induced LCO and internal resonance, but the results make it

11


clear that nonlinear analysis is necessary to model the effect of the store. The study notes

the difficultly of studying internal resonance in a highly damped systems because amplitudes

drop too quickly to identify amplitude modulation.

Sheta et al.[22] analyzed the NATA model with a high fidelity multidisciplinary com-

putational environment (MDICE). A nonlinear structure typical of the previous studies of

the NATA model was used. Aerodynamics were computed with the full Navier-Stokes equa-

tions, which are generally nonlinear, using CFD-Fastran. MDICE was used for aeroelastic

simulation. The response to an initial plunge displacement was studied. The predicted pitch

and plunge response agreed well with experimental observations over a range of freestream

velocities. As expected, LCO amplitude increased with the freestream velocity. Theory and

experiment showed a small change in frequency over velocity range. The relatively small

variation in LCO behavior over the large freestream velocity range used suggests that the

LCO was strongly dependent on the structural nonlinearity. The study demonstrated the

importance of modeling the flow viscosity by comparing the MDICE results to a simpler

aerodynamic model. The authors claim the viscosity effects can not be modeled accurately

if analytic aerodynamic formulations are used and that the accurate results show the impor-

tance of a high fidelity model.

2.2.2 Continuous Systems

Beam-like Cantilever Wings

Kim and Strganac[23] studied the effect of various nonlinearities on the LCO behavior of a

cantilevered beam-like wing. Structural nonlinearities were included to the third order with

nonlinear beam equations. Aerodynamic nonlinearities were included with a quasi-steady

model including a simple cubic stall model. Kinematic nonlinearities were included for a

store attached beneath the wing. The Galerkin Method was used to transform the PDEs

into ODEs which were then converted to state space form. Phase portraits were used to

investigate the effect of each nonlinearity individually and in all possible combinations. All

cases were examined above and below the linear flutter velocity. The motion damped out be-

12


low the flutter velocity and continuously increases in amplitude above the flutter velocity for

nonlinearities in structure only, store only, and structure and store. For aerodynamic nonlin-

earity alone and aerodynamic with structural or store nonlinearities, the motion damped out

below the flutter velocity and goes to a single LCO above the flutter velocity. When all three

nonlinearities were combined, a single LCO is found above the flutter velocity. Below the

flutter velocity, the behavior depended on the initial condition. For a small initial condition,

the motion damped out, but for a larger one the motion went to an LCO.

In a follow up study, Kim and Strganac[24] included variation of the mass, inertia,

and position of the under wing store. The LCO behavior was found to be dependent on the

store parameters. In all the cases considered, LCO existed below the linear flutter speed for

a larger initial condition and motion damped out for a smaller initial condition. Otherwise

the results were similar to the previous study. Poincare maps were also used to help better

understand the bifurcation behavior.

Plate-like Wings

Tang et al.[25] studied LCO of a cantilevered rectangular plate wing. A three dimensional,

time domain, unsteady vortex lattice model with a reduced-order aerodynamic eigenmode

technique was used to model the aerodynamics. A quasi-static correction was used to account

for the neglected aerodynamic eigenmodes. Nonlinear structural equations were derived with

Hamilton’s Principle and Lagrange’s Equations based on the Von Karman nonlinear plate

equations. Approximate structural modes were used to with Lagrange’s equations. The

method assumed that all nonconservative forces acted in the direction perpendicular to the

plate. The system was written in discrete time form and a standard discrete time algorithm

was used to calculate the nonlinear response of the system with the complete and reduced

order aerodynamic model. It was shown that only a few aerodynamic eigenmodes need to

be retained in the aeroelastic model to attain good accuracy. The plate wing was a constant

thickness with aspect ratio varied from 0.75 to 10 for different cases. The linear flutter

motion was shown to be dominated by the coupling between the first two structural modes,

13


spanwise bending and the rigid plunge and rotation modes in the chordwise direction. With

the velocity set at less than linear flutter velocity, the motion damped out. When the velocity

was set greater than the linear flutter velocity an LCO was shown to exist. There was good

agreement between the full and reduced aerodynamic model results. The LCO frequency

and amplitude were found to increase as the flow velocity increases. The amplitude increased

faster when the aspect ratio was higher.

Tang et al.[26] went on to study the LCO of a cantilevered plate delta wing. The

aerodynamic and structural models followed the same procedure as the previous except that

they were adapted to the delta wing shape. Linear flutter results were shown to be nearly

identical for the full and reduced order aerodynamic models. Results were very close between

the reduced and full model in nonlinear cases. An experiment was conducted with a 45 degree

Lucite wing with center 60% of root clamped. Good agreement was found between theoretical

and experimental natural frequencies. Good agreement was also found between the theory

and experimental LCO amplitudes and frequencies.

Tang and Dowell[27] expanded the previous study of LCO in a delta wing model to

study the effect of an initial steady angle of attack. The same theoretical model was used

with the addition of a steady angle of attack term. The initial angle of attack was varied

from 0 to 2 degrees. The authors discussed how the structural natural frequency varied as

the airspeed increased due to increased static deflection. The analysis showed that as the

static angle of attack was increased the flutter velocity decreased. At higher static angles the

primary mode of flutter changed from the coupling of the first structural modes to coupling of

higher order modes. LCO was only observed above the linear flutter velocity for this system.

As the flow velocity increased, both the static and dynamic amplitude were seen to increase

with a small angle of attack. After crossing a certain angle, the dynamic amplitude was

seen to decrease as the velocity was increased further. At even higher angles of attack, the

dynamic amplitude was shown to completely disappear. At one of the higher angles of attack,

the LCO amplitude and frequency experienced a jump phenomenon from a larger dynamic

amplitude to a small one. The frequency increased more that two-fold at the same jump

14


point. At the next angle, the LCO was dominated by the after jump behavior. Overall the

maximum LCO amplitude is shown to decrease as the steady angle of attack is increased for

a given speed. The frequency rose and then fell slowly before the jump to higher frequency,

after which it stayed relatively constant as the steady alpha is increased further. The jump

is supposed to correspond to the change from the lower to higher structural modes.

Attar et al.[28] expanded the steady angle of attack study to include experimental

data. In the experiment the static angle of attack was set and then the airspeed was incre-

mentally increased. Five cases were examined from 0 to 4 degrees in 1 degree increments.

Theoretically, both clamped and free restraints on the in-plane motion were considered. The

theoretical and experimental plate natural frequencies matched well. The fully clamped in-

plane boundary condition proved to be much too stiff and the zero restraint case modeled

the experimental behavior better. The velocity for this study was limited to a narrow band.

All the LCO found consisted of a coupling between the first bending and first torsion modes.

The theoretical model showed a slight increase in frequency with both angle of attack and

flow velocity while the experiment showed a slight decrease. Both theoretical and experimen-

tal frequency changes were very small. Both the experiment and theory showed an increase

followed by a decrease in the flutter speed as the static angle of attack increased. Both also

showed an increase in the magnitude of the LCO at a given velocity for higher static angle

of attack. The qualitative results agreed relatively well.

The same delta wing model was used yet again by Tang et al.[29]. For this study a

store was added near the wing tip. The store was modeled as a slender rigid body attached

to the wing at two points. The forward connection was pinned and the aft connection was

a spring. A slender body aerodynamic model was applied to the store. The effect of the

placement of the store on the wing was studied and it was found that the LCO behavior was

sensitive to the store span location and the stiffness of the spring connection. When the store

was near the leading edge, the flutter velocity was higher than if the store was to the rear.

The theory predicted the flutter velocity and the frequency well, but it did not accurately

predict the amplitude of the LCO.

15


Tang and Dowell[30] added a freeplay gap to the spring connection at the store’s aft

end. For this study, the store was moved around the wing and the freeplay gap was varied.

The LCO amplitude had a significant dependence on the store’s initial conditions and pitch

motion. As in the previous study, the LCO was sensitive to the location of the store and

additionally it was sensitive to the freeplay gap value. The theoretical results were good

around the flutter velocity in a qualitative sense, but were poor at higher velocities.

Attar, Dowell, and White[31] and Attar, Dowell, and Tang[32] used a high fidelity

nonlinear structural model with a linear vortex lattice model in the finite element package

ANSYS to study the delta wing model from the previous studies with and without a store. For

cases without the store, the correlation between the high fidelity model and the experiment

were better than the von Karman model. The LCO amplitudes and frequencies predicted

by the high fidelity model matched well with the experimental values. When the store was

added to the high fidelity model, it accurately predicted sensitivity to the spanwise location

of the store. With the store placed closer to the root, the experiment and theory showed that

the model had a low sensitivity to an initial non-zero angle of attack. With the store placed

further out the wing, the experimental model showed a high sensitivity of the flutter velocity

to an initial angle of attack. This behavior was not predicted by the analytical model and it

was suspected that the simplicity of the aerodynamic model was to blame.

Attar and Gordnier[33] analyzed a cropped delta wing model with a well validated

finite difference Euler fluid solver implicitly coupled to a high fidelity finite element structural

solver via subiteration. The structural solver was formulated in a co-rotational manner to

accurately model large deflections and rotations. This was an improvement over von Karman

theory which allows only for moderate rotations. The solver did not significantly improve

over previous simpler formulations in the flow regime where experimental data was available;

however, it differed from the simpler model as the flow velocity was increased more and

the LCO amplitude became higher. The current model was thought to be a more accurate

prediction because it was better able to capture the aerodynamic and structural effects at

the higher amplitudes.

16


2.3 Nonlinear Normal Modes

Shaw and Pierre[34] extended the linear system concept of normal modes of vibration to

general nonlinear systems. Drawing inspiration from the center manifold reduction technique,

they constructed invariant manifolds which represent a normal mode of motion for nonlinear

systems with finite degrees of freedom. The initial paper constructed the manifold with

an approximate asymptotic approach. This approach limits itself to systems with weak

nonlinearities and the order of accuracy cannot be determined a priori.

A number of extensions and improvements have been made to the method of nonlinear

normal modes. Shaw and Pierre[35] reformulated the method in terms of continuous systems.

The original single mode manifolds were incapable of modeling internal resonances present

in the system. Pesheck et al.[36] developed multi-mode manifolds which made it possible to

include coupling between modes due to internal resonance in a single nonlinear normal mode.

A Galerkin-based approach of formulating the manifolds was introduced which allowed a user

to prescribe a domain and the order of accuracy of the nonlinear normal mode[37]. All of the

nonlinear normal modes mentioned so far assumed the system was not forced. Jiang, Pierre,

and Shaw[38] incorporated the ability to include a harmonic excitation into the method.

The application of nonlinear normal modes to structural dynamics problems was

considered by Pierre et al[39]. The method was demonstrated with the Galerkin formulation

on the classic spring-mass-damper problem including nonlinearities, a finite element beam

supported by a nonlinear spring on one end, and on a simple finite element rotating blade

model. The frequency response of the beam model was also shown with the method including

harmonic excitation. Pesheck et al.[40] presented a thorough study of nonlinear normal modes

applied the structural dynamics of a helicopter rotor blade model.

17

Chapter 3

Nonlinear Normal Modes

Nonlinear normal modes (NNM) are an extention of the theory of linear normal modes to

systems with nonlinearities. A modal motion relationship is formulated through the use

of invariant manifolds. In this chapter the theory of NNM is developed. The second-order

formulation of Shaw and Pierre[34] is presented first and includes an example. This derivation

includes the asymptotic manifold approximation. Then, a generalization of the second-

order formulation to include first-order systems is presented. The first-order formulation was

required to allow the use of unsteady aerodynamics and certian master coordinates useful in

the analysis of aeroelastic LCO. The final two sections detail new approaches for calculating

the modal coefficients solution, including the Galerkin approach and a collocation approach.

The collocation approach is ideal for the LCO studied in this dissertation.

3.1 Second-Order Formulation

with Asymptotic Modal Coefficient Solution

This section mostly follows the derivation of nonlinear normal modes by Shaw and Pierre[34].

A few changes in notation and description have been made to aid in a quick understanding

of the method.

The derivation of nonlinear normal modes begins with an original system ofN coupled,

18

Chapter 3. Nonlinear Normal Modes

second-order differential equations. They must be written in the form of Eq. (3.1) or must

be transformed to this form.

xi = fi [x1, x2, ..., xN , x1, x2, ..., xN ] (3.1)

The N second-order equations are then transformed to 2N first-order differential equations.

xi = yi

yi = fi [x1, x2, ..., xN , y1, y2, ..., yN ](3.2)

One of the xi-yi coordinate pairs must be chosen as the master coordinate on which

the nonlinear normal modes are constructed. Pair one is chosen for convenience of notation,

but any of the pairs can be chosen.

x1 = u

y1 = v(3.3)

The modal degrees of freedom are denoted here as u and v. The remaining system degrees

of freedom are slaved to the master modal coordinates by a rule for modal motion.

xi = Xi[u, v]

yi = Yi[u, v](3.4)

In the case of linear normal modes, only a linear relationship would exist between the master

and slave coordinates; however, for nonlinear normal modes, the realationship is considered

to be of a general nonlinear form. To create an asymptotic approximation of the nonlinear

normal modes, Xi[u, v] and Yi[u, v] are expanded as a Taylor series in two variables to the

19


desired order of the approximation.

Xi[u, v] = a1iu+ a2iv + a3iu2 + a4iuv + a5iv

2 + ...

Yi[u, v] = b1iu+ b2iv + b3iu2 + b4iuv + b5iv

2 + ...(3.5)

The a’s and b’s are unknown coefficients. The first step in solving for the unknown coefficients

is to take the derivative of Eq. (3.4).

xi =∂Xi

∂uu+

∂Xi

∂vv

yi =∂Yi∂u

u+∂Yi∂v

v

(3.6)

The equations of motion, Eq. (3.2), are used to substitute for xi and yi.

yi =∂Xi

∂uu+

∂Xi

∂vv

fi [x1, x2, ..., xN , y1, y2, ..., yN ] =∂Yi∂u

u+∂Yi∂v

v

(3.7)

Since u = x1 = y1 = v and v = y1 = f1[x1, ..., xN , y1, ..., yN ], Eq. (3.7) can be rewritten as

yi =∂Xi

∂uv +

∂Xi

∂vf1 [x1, x2, ..., xN , y1, y2, ..., yN ]

fi [x1, x2, ..., xN , y1, y2, ..., yN ] =∂Yi∂u

v +∂Yi∂v

f1 [x1, x2, ..., xN , y1, y2, ..., yN ] .

(3.8)

The remaining x1’s and y1’s are replaced by u and v and the xi’s and yi’s are replaced by

Xi[u, v] and Yi[u, v].

Yi[u, v] =∂Xi

∂uv +

∂Xi

∂vf1 [u,X2[u, v], ..., XN [u, v], v, Y2[u, v], ..., YN [u, v]]

fi [u,X2[u, v], ..., XN [u, v], v, Y2[u, v], ..., YN [u, v]]

=∂Yi∂u

v +∂Yi∂v

f1 [u,X2[u, v], ..., XN [u, v], v, Y2[u, v], ..., YN [u, v]]

(3.9)

The above equations are polynomials in u and v with coefficients composed of combinations

20


of the unknown a and b coefficients from Eq. (3.5). The form is like

zo∑i=0

(zo)−i∑j=0

Cijuivj = 0, C00 = 0. (3.10)

Here z is the order of the nonlinear modal approximation in Eq. (3.5) and o the order of the

nonlinearities of the system from Eq. (3.1). The Cij terms are composed of unknown a and

b coefficients. At this point, one of several options may be chosen to solve for the unknown

coefficients. For this derivation, the asymptotic method is used because it is the simplest

and most common. New solution methods were developed and will be presented later in this

chapter.

For the asymptotic method, the only terms from Eq. (3.10) retained are ones in which

(i + j) is less than or equal to the order of the modal approximation z. The higher order

terms are not accounted for in the solution leading to the possiblity of inaccuracies.

z∑i=0

z−i∑j=0

Cijuivj +H.O.T. = 0, C00 = 0 (3.11)

The Cij’s are extracted and set equal to zero creating a system of equations in a and b.

Cij = 0, (i+ j) ≤ z (3.12)

This system is nonlinear in the unknown a and b coefficients and gives multiple solutions.

The coefficients can be solved for in groups associated with the combined polynomial order

(i+j). Referring back to Eq. (3.5), the linear coefficients a1i, a2i, b1i, and b2i can be solved as

a set followed by the quadratic coefficients a3i, a4i, a5i, b3i, b4i, and b5i, and so on. The linear

order equations have some real and some complex solutions. Each real solution corresponds

to one of the nonlinear modal modes of motion. The complex solutions are meaningless and

disregarded. The real solutions can also be obtained from a complex eigenvalue analysis

performed on a linearized version of the original system. The eigenvalue method of solving

21


for these linear coefficients will be shown in the spring-mass-damper example below. Each

real solution obtained either from the solution to the linear order equations or the eigenvalue

analysis can then be substituted into the higher order coefficient equations. After the substi-

tution, the higher order equations are linear in the unknown coefficients and can be solved

easily.

The modes generated with the coefficient solution to the linear order equations are

equivalent to the modes obtained from a complex eigenvalue analysis performed on a lin-

earized version of the original system. The eigenvalue method of solving for these linear

coefficients will be shown in the spring-mass-damper example below.

With the unknown coefficients now known, the modal rules can be inserted into the

equations of motion of the master coordinate. Each set of a and b coefficients create a

different nonlinear normal mode. The modal form of the equation will appear in first-order

oscillator form as

u = v

v = f1 [u,X2[u, v], ..., XN [u, v], v, Y2[u, v], ..., YN [u, v]](3.13)

or in second-order form

u = f1 [u,X2 [u, u] , ..., XN [u, u] , u, Y2 [u, u] , ..., YN [u, u]] . (3.14)

This equation can then be used to find the systems behavior in that mode. The modal

motion found can be converted back into physical coordinates if desired by using Eqs. (3.3)

and (3.4).

3.1.1 Linear Spring-Mass-Damper Example

This section applies the nonlinear normal mode method to a linear three degree of freedom

spring-mass-damper problem to demonstrate the method and aid in understanding. It follows

the process described in the previous section. The system to be used can be seen in Fig. 3.1.

22


Figure 3.1: Three Degree-of-Freedom Spring-Mass-Damper System

The equations of motion for this system are

mx1 = −cx1 + kx2 − 2kx1

mx2 = kx3 − 2kx2 + kx1

mx3 = −kx3 + kx2.

(3.15)

In these equations, m is the mass of each block, k is the linear spring stiffness of each spring,

and c is the damping coefficient. These equations are then converted to first-order form.

x1 = y1

y1 = f1 [x1, x2, x3, y1, y2, y3] = −cy1 + kx2 − 2kx1

x2 = y2

y2 = f2 [x1, x2, x3, y1, y2, y3] = kx3 − 2kx2 + kx1

x3 = y3

y3 = f3 [x1, x2, x3, y1, y2, y3] = −kx3 + kx2

(3.16)

23


The master coordinate is chosen to be x1-y1 and the modal motion rules are formed. Since

this is a linear system, only the linear expansion is used.

x1 = u

y1 = v(3.17)

x2 = X2[u, v] = a1u+ a2v

y2 = Y2[u, v] = b1u+ b2v

x3 = X3[u, v] = c1u+ c2v

y3 = Y3[u, v] = d1u+ d2v

(3.18)

The solution for the unknown a, b, c, and d coefficients must be found. The first step is to

take the time derivative of Eqs. (3.18).

X2 = a1u+ a2v

Y2 = b1u+ b2v

X3 = c1u+ c2v

Y3 = d1u+ d2v

(3.19)

Substitutions are now made according to Eqs. (3.7), (3.8), and (3.9) to get

b1u+ (−a1 + b2) v − a22kv + a2 (2ku− a1ku+ cv) = 0

− b1v + b2cv + k ((1 + 2b2 − a1 (2 + b2) + c1)u+ (−a2 (2 + b2) + c2) v) = 0

d1u+ (−c1 + d2) v + c2 (cv − k ((−2 + a1)u+ a2v)) = 0

− c1ku− a1 (−1 + d2) ku+ 2d2ku− d1v + cd2v + a2kv − c2kv − a2d2kv = 0.

(3.20)

24


The coefficients of u and v in each of the four equations are extracted and set equal to zero

to form the system below.

2ka2 − ka1a2 + b1 = 0

k − 2ka1 + 2kb2 − ka1b2 + kc1 = 0

b1 + 2kc2 − ka1c2 = 0

ka1 + 2kb2 − ka1b2 − kc1 = 0

− a1 + ca2 − ka22 + b2 = 0

− 2ka2 − b1 + cb2 − ka2b2 + kc2 = 0

b2 − c1 + cc2 − ka2c2 = 0

ka2 − b1 + cb2 − ka2b2 − kc2 = 0

(3.21)

The system coefficients must be calculated numerically to solve this system. For simplic-

ity, the mass of each block and all the spring stiffnesses are taken as unity. The damping

coefficient is set to 0.3.

2a2 − a1a2 + b1 = 0

1− 2a1 + 2b2 − a1b2 + c1 = 0

b1 + 2c2 − a1c2 = 0

a1 + 2b2 − a1b2 − c1 = 0

− a1 + 0.3a2 − a22 + b2 = 0

− 2a2 − b1 + 0.3b2 − a2b2 + c2 = 0

b2 − c1 + 0.3c2 − a2c2 = 0

a2 − b1 + 0.3b2 − a2b2 − c2 = 0

(3.22)

The solution of these equations results in three sets of real coefficients and twelve complex

solutions. The complex solutions are meaningless in the terms the problem was formulated

and are discarded. The remaining three sets of real coefficients determine the modes of

25


motion. The modes created with these coefficients are identical to the modes found through

an eigenvalue analysis. The three sets of modal coefficients can be seen below.

Mode 1 Mode 2 Mode 3

a1 = −1.211 0.435 1.801

a2 = 0.1982 0.1341 0.268

b1 = −0.636 −0.210 −0.0533

b2 = −1.231 0.413 1.792

c1 = 0.531 −0.776 2.25

c2 = −0.1146 −0.01014 0.424

d1 = 0.368 0.01586 −0.0844

d2 = 0.542 −0.774 2.23

(3.23)

The modal rules can now be fully defined for each mode of motion. Any mode calculated

above can be substituted into the first equation of motion to derive the corresponding modal

equation. The subscripts in Eq. (3.24) refer to the modal degree of freedom.

u1 = v1

v1 = −3.21u1 − 0.1018v1

u2 = v2

v2 = −1.565u2 − 0.1659v2

u3 = v3

v3 = −0.1991u3 − 0.0323v3

(3.24)

In second-order form, these equations are

u1 = −3.21u1 − 0.1018u1

u2 = −1.565u2 − 0.1659u2

u3 = −0.1991u3 − 0.0323u3.

(3.25)

26


This is exactly the same result that is found by traditional linear modal decomposition.

These modal equations can now be simulated or otherwise analyzed to determine the system’s

behavior in the modes.

3.1.2 Alternative Eigenvalue Solution to Linear Coefficients

When using the asymptotic method to solve for the unknown modal coefficients, the equations

used to solve for the linear coefficients are nonlinear. These equations can be solved for a

simple system in a short time, but as the complexity of the system increases the time required

to solve these equations grows rapidly. For example, with three bending and three torsion

mode shapes for the beam model presented later, the nonlinear equation solver included

in Mathematica did not give the solution within two hours. However, the linear modal

coefficients can be found with an eigenvalue solution of the linearized system. The eigenvalue

solution to the linear coefficients is also used as the first step in the Galerkin and collocation

solution methods presented in this dissertation.

Starting with the linearized system written in second-order matrix form

[M ] {qi}+ [C] {qi}+ [K] {qi} = 0, (3.26)

the first-order stiffness matrix is formed, where [0] and [I] are zero and identity matrices of

the correct size.

A =

[0] [I]

−M−1K −M−1C

(3.27)

For a first-order system, the A matrix would be known directly. The eigenvectors of A are

found and written column wise defining a modal matrix Z. Each column (eigenvector) of

Z is normalized by the component corresponding to the desired master coordinate creating

Z∗. From here, a transformation matrix is desired that would transform the original system

matrix A into block diagonal of oscillator equations. This is achieved through a matrix W

27


which is composed of the complex conjugate pairs of the eigenvalues λi and λ∗i .

W = BlockDiagonal

...

1 1

λi λ∗i

...

(3.28)

The transformation matrix U = Z∗W−1 will transform the A matrix to the block diagonal

first-order oscillator form just like the first-order modal equation, Eq. (3.13), in the NNM

derivation. Each set of two columns in the matrix U corresponds to a modal solution.

The three degree of freedom, spring-mass-damper system from Fig.3.1 and Eq. (3.15)

is used here as an example of this process. In second-order matrix form Eq. (3.15) becomes

1 0 0

0 1 0

0 0 1

x1

x2

x3

+

0.3 0 0

0 0 0

0 0 0

x1

x2

x3

+

2 −1 0

−1 2 −1

0 −1 1

x1

x2

x3

=

0

0

0

.

(3.29)

The first-order system matrix is then easily computed.

A =

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

−2 1 0 −0.3 0 0

1 −2 1 0 0 0

0 1 −1 0 0 0

(3.30)

Finding the eigenvectors of A, the Z matrix is created and normalized by the row corre-sponding to the first degree-of-freedom which is being used as the master coordinate.

Z∗

=

1 1 1 1 1 1

−1.221 + 0.355i −1.221 − 0.355i 0.424 + 0.1674i 0.424 − 0.1674i 1.797 + 0.1194i 1.797 − 0.1194i

0.537 − 0.205i 0.537 + 0.205i −0.775 − 0.01266i −0.775 + 0.01266i 2.24 + 0.1891i 2.24 − 0.1891i

−0.0509 + 1.791i −0.0509 − 1.791i −0.0829 + 1.248i −0.0829 − 1.248i −0.01615 + 0.446i −0.01615 − 0.446i

−0.574 − 2.20i −0.574 + 2.20i −0.244 + 0.516i −0.244 − 0.516i −0.0822 + 0.799i −0.0822 − 0.799i

0.340 + 0.971i 0.340 − 0.971i 0.0801 − 0.966i 0.0801 + 0.966i −0.1205 + 0.995i −0.1205 − 0.995i

(3.31)

28


The matrix W is then formulated using the complex conjugate pairs of eigenvalues.

W =

1 1 0 0 0 0

−0.0509 + 1.791i −0.0509 − 1.791i 0 0 0 0

0 0 1 1 0 0

0 0 −0.0829 + 1.248i −0.0829 − 1.248i 0 0

0 0 0 0 1 1

0 0 0 0 −0.01615 + 0.446i −0.01615 − 0.446i

(3.32)

Finally the transformation matrix, U is found.

U =

1 0 1 0 1 0

−1.211 0.1982 0.435 0.1341 1.801 0.268

0.531 −0.1146 −0.776 −0.01014 2.25 0.424

0 1 0 1 0 1

−0.636 −1.231 −0.210 0.413 −0.0533 1.792

0.368 0.542 0.01586 −0.774 −0.0844 2.23

(3.33)

If the first-order system matrix is transformed with this matrix, U−1AU , the first-order

oscillator form would result.

A∗ =

0 1 0 0 0 0

−3.21 −0.1018 0 0 0 0

0 0 0 1 0 0

0 0 −1.565 −0.1659 0 0

0 0 0 0 0 1

0 0 0 0 −0.1991 −0.0323

(3.34)

Now referring back to the modal definition equation from the NNM derivation, Eq. (3.5), it

29


can be deciphered that the matrix U is of the form

U =

1 0 1 0 1 0

a1,1 a2,1 a1,2 a2,2 a1,3 a2,3

c1,1 c2,1 c1,2 c2,2 c1,3 c2,3

0 1 0 1 0 1

b1,1 b2,1 b1,2 b2,2 b1,3 b2,3

d1,1 d2,1 d1,2 d2,2 d1,3 d2,3

, (3.35)

where the second index on the coefficients corresponds to the mode. Referring back to the

previous example, the pairs of columns in the U transformation matrix correspond to three

modal solutions seen in Eq. (3.23).

3.1.3 Linear Normal Mode (LNM) Solution

Throughout this dissertation the results for nonlinear normal mode (NNM) solutions are

compared to a linear normal mode (LNM) solution. The LNM solution is a simplification of

the NNM solution. The nonlinear modal approximation function from Eq. (3.5) is simplified

to a linear modal approximation function.

Xi[u, v] = a1iu+ a2iv

Yi[u, v] = b1iu+ b2iv(3.36)

The rest of the process for generating the NNM is followed with this simplified modal ap-

proximation. This results in a model that captures the nonlinearity in the master coordinate,

but with only linear coupling between the modes.

30


3.2 First-Order Formulation

Two systems encountered in the work for this dissertation could not be transformed to

conform to the standard formulation of nonlinear normal modes currently in literature. In

the first instance, using unsteady aerodynamics with the pitch-plunge airfoil created a system

with two second-order and two first-order differential equations. Obviously, this system could

be transformed into six first-order differential equations, but cannot be written in a second-

order form. The second instance was discovered during a study on the effect of the master

coordinate. When the linear flutter mode was chosen as the master coordinate for the

pitch-plunge airfoil, the system consisted of four coupled, first-order differential equations

that could not be transformed back into two coupled, second-order differential equations.

Published methods only allow for a set of second-order differential equations, Eq. (3.1).

Encountering these two cases necessitated deriving a formulation of NNM for a system of

coupled, first-order equations. As such, this derivation starts with a system of the form

xi = fi [x1, x2, ..., xN ] . (3.37)

Two master coordinates are chosen and the remaining coordinates are slaved to the master

coordinates.

xj = u

xk = v(3.38)

xi = Xi[u, v] i 6= j, k (3.39)

A two variable Taylor series is then assumed for the modal relationship with unknown coef-

ficients ami, where m is the number of terms in the Taylor series.

Xi[u, v] = a1iu+ a2iv + a3iu2 + a4iuv + a5iv

2 + ... i 6= j, k (3.40)

31


To solve for the unknown coefficients, the time derivative of Eq. (3.39) is taken.

xi =∂Xi

∂uu+

∂Xi

∂vv i 6= j, k (3.41)

Using Eq. (3.38), xj and xk are substituted for u and v, respectively.

xi =∂Xi

∂uxj +

∂Xi

∂vxk i 6= j, k (3.42)

Then using the equations of motion, Eq. (3.37), xi, xj, and xk are replaced.

fi [x1, ..., xN ] =∂Xi

∂ufj [x1, ..., xN ] +

∂Xi

∂vfk [x1, ..., xN ] i 6= j, k (3.43)

Finally, the rules for modal motion, Eqs. (3.39) and (3.40), are used to get the final form of

the equations.

fi [u, v,Xi[u, v], ..., Xn[u, v]]

−(∂Xi[u, v]

∂ufj [u, v,Xi[u, v], ..., Xn[u, v]] +

∂Xi[u, v]

∂vfk [u, v,Xi[u, v], ..., Xn[u, v]]

)= 0

i 6= j, k

(3.44)

The unknown coefficients can be found from these equations with the asymptotic method as

previously presented. Eq. (3.44) can also be treated as a residual equation for the approximate

nonlinear normal mode and a weighted residual type method can be used to solve for the

coefficients. Two such methods are developed in the next two sections.

3.3 Galerkin Modal Coefficient Solution

The asymptotic solution method for the unknown modal coefficients presented as part of the

second-order NNM derivation is only accurate for weakly nonlinear systems and a limited

32


amplitude range. The asymptotic method cannot be trusted for a system with a relatively

large amplitude. In fact, if the amplitude grows too large, the asymptotically generated

modes will become unstable. The unstable modes exhibit a divergent solution and fail to

capture LCO. In aeroelastic LCO, the amplitude of the motion increases when the veloc-

ity is increased which creates a problem with the asymptotic solution at higher velocities.

Additionally, no method exists for determining the error of the asymptotic method without

knowing the exact solution and increasing the order of the approximation function will not

necessarily improve the results[34][37]. In an attempt to alleviate these deficiencies in the

asymptotic method and improve the results obtained herein, a Galerkin based solution was

attempted. The Galerkin method is a weighted residual method that drives a weighted aver-

age of the approximation function’s error to zero over a specified domain. Pesheck et al.[37]

transformed their system into an amplitude and phase domain to formulate a Galerkin based

solution to the modal coefficients. A combination polynomial and transcendental modal rela-

tion was used and good results were obtained. A simpler, more direct version of the Galerkin

solution was attempted here which applied the Galerkin method in the same domain and

with the same modal relationship as the asymptotic method. Remaining in the same domain

facilitated comparisons between the different coefficient solution methods.

The process is presented here using the second-order equations, but an exact parallel

exists for Eq. (3.44) in the first-order derivation. Starting with Eq. (3.9) from the second-

order derivation, all terms are moved to one side and the result is treated as a residual

equation.

R1i = Yi(u, v)−(∂Xi

∂uv +

∂Xi

∂vf1 (u,X2(u, v), ..., XN(u, v), v, Y2(u, v), ..., YN(u, v))

)R2i = fi (u,X2(u, v), ..., XN(u, v), v, Y2(u, v), ..., YN(u, v))

−(∂Yi∂u

v +∂Yi∂v

f1 (u,X2(u, v), ..., XN(u, v), v, Y2(u, v), ..., YN(u, v))

) (3.45)

The Galerkin approach has the advantage that all the terms in this equation can be considered

as it does not have to be truncated at the order of the approximation as in the asymptotic

33


method.

The modal expansions from Eq. (3.5) are considered to be of the form

Xi[u, v] =n∑j=1

aj,iφj[u, v]

Yi[u, v] =n∑j=1

bj,iφj[u, v].

(3.46)

Here the φj[u, v] shape functions could be chosen as any appropriate set. For this study,

the shapes were kept identical to the polynomial set used in the asymptotic formulation for

an easier and more direct comparison between the asymptotic and Galerkin solutions. Note

that the first two coefficients in each expansion correspond to the linear order solution and

can be found exactly by the eigenvalue method. Consequently, there are two choices: 1)

the Galerkin integral can be used to solve for all the coefficients; or 2) the linear coefficients

can be found with the eigenvalue method and then the Galerkin integral can be used to

solve for just the coefficients of nonlinear order. Using the eigenvalue method has two main

advantages. The linear portion of the solution will be exact and the solution of the coefficients

from the result of the Galerkin integrals will be much simpler. Given these advantages, the

eigenvalue-Galerkin combination was always used here. As such, the Galerkin integrals are

formed where SUV is the region of interest for the u and v modal variables and n is the

number of terms in the modal approximation function.

G1,j =

∫ ∫SUV

φj[u, v]R1,jdudv

G2,j =

∫ ∫SUV

φj[u, v]R2,jdudv, for j = {3, 4, ..., n}(3.47)

The region SUV must be selected appropriately. If the region is too small, the LCO may

lie outside the region, providing no guarantee of accuracy. On the other hand, if it is too

large, the accuaracy of the LCO solution will degrade due to the large area over which

the error must be averaged. To pick a good region size, the asymptotic solution was used

34


to guess the motion’s amplitude. The equations resulting from the Galerkin integrals are

used to solve for the unknown nonlinear coefficients; however, unlike the asymptotic method

these equations are nonlinear and computationally intensive to both derive and solve. Even

for the quasi-steady pitch-plunge airfoil system, the Galerkin method took several orders of

magnitude more time than the asymptotic method to arrive at a solution for the coefficients,

558 vs. 0.172 seconds for each nonlinear normal mode. Additionally, in Pesheck et al.[37],

a supercomputer was necessary to solve for the modal coefficients in their version of the

Galerkin solution. This method was used to solve some problems; however, more work in the

computational implementation is needed for the Galerkin method to be applicable to real or

even relativly simple problems.

3.4 Collocation Modal Coefficient Solution

The best method for solving for the modal coefficients attempted was the collocation method.

Like the Galerkin method, the collocation solution considers all the terms from the resid-

ual equation, the error can be reduced by increasing the order of the approximation, and

the modes generated remain stable even at very large amplitudes, filling deficiencies of the

asymptotic method. Both the Galerkin and collocation solutions are weighted residual meth-

ods. The Galerkin method uses the shape functions as the weights while the collocation uses

Dirac’s delta function. While the Galerkin method sets the weighted average residual over

a region of interest equal zero, the collocation method forces the residual to zero at specific

points in the domain. This is ideally suited to steady state LCO since it occurs on a single

path where accuracy is desired and points can be selected. Away from LCO path the ac-

curacy of the solution is unimportant unless accurate information about the growth of the

motion is desired.

The eigenvalue method is again used for the solution to the linear coefficients. A set

of points where solution accuray is desired must be chosen in the u and v domain. When the

dirac delta function is used as the weight in the residual integral, the result simply substitutes

35


this chosen set of ui and vi points into the residual equations, Eq. (3.45) (or Eq. (3.44) for

the first-order version).

R1,i,j = Yi (uj, vj)−(∂Xi

∂uvj +

∂Xi

∂vf1 (uj, X2 (uj, vj) , ..., XN (uj, vj) , vj, Y2 (uj, vj) , ..., YN (uj, vj))

)R2,i,j = fi (uj, X2 (uj, vj) , ..., XN (uj, vj) , vj, Y2 (uj, vj) , ..., YN (uj, vj))

−(∂Yi∂u

vj +∂Yi∂v

f1 (uj, X2 (uj, vj) , ..., XN (uj, vj) , vj, Y2 (uj, vj) , ..., YN (uj, vj))

)(3.48)

These equations can then be solved for the remaining nonlinear modal coefficients. The

solution to these equations take on the same order of time as the asymptotic solution, both

of which are much faster than the Galerkin solution

The main difficulty with the collocation method, especially in two-dimensions as it

must be used here, is the selection of the points where the residual is forced to zero. Based

on the type of nonlinearities, certain symmetries may be exploited. Through a study of the

form of the residual equations, Eq. (3.45), it was found that for cubic stiffening the first and

third quadrants are antisymmetric and the second and fourth quadrants are antisymmetric.

As such, points chosen in the first and second quadrants will force the corresponding points

in the other quadrants to be zero as well. However, there is still not a good way to pick

the points since the amplitude and shape of the motion are not known a priori. Since the

amplitude and shape must be known to select a good set of points, an asymptotic solution

was completed first. A set of equally spaced points in two adjacent quadrants was selected

from the asymptotic solution for the ui and vi points neccessary for collocation solution.

36

Chapter 4

Physical Models

This chapter provides details on the physical models considered in this work. To achieve a

series of objectives, four different models were considered. The first model was a spring-mass-

damper (SMD) system with two degrees of freedom and a nonlinear Van der Pol oscillator

style damping term[41]. Since limit cycle oscillation (LCO) had not previously been modeled

with nonlinear normal modes (NNM), the SMD model was used as an initial test to prove

that the NNM method was capable of capturing LCO. Additionally, the system’s parameters

were varied to study the effect on NNM solution accuracy. The second and most frequently

utilized model was a nonlinear variation of the classic pitch-plunge airfoil model that appears

extensively in aeroelasticity literature. This model had quasi-steady aerodynamics and a

nonlinear pitch stiffness. The quasi-steady, pitch-plunge model was used to accomplish many

objectives of the dissertation including modeling of aeroelastic LCO with NNM, studying

the effect of the NNM’s master coordinate, and accuracy improvements of multiple modal

coefficient solution methods. The same pitch-plunge airfoil structure was also coupled with

an unsteady aerodynamic model. The unsteady version’s main purpose was to adapt the

NNM process to deal with the addition of nonstructural, first-order aerodynamic degrees of

freedom. The final model considered was an adaptation of the Goland wing model. The

Goland wing was coupled with a nonlinear, quasi-steady aerodynamic model. The beam

model chiefly allowed the exploration of a more realistic system with many degrees of freedom

37

Chapter 4. Physical Models

and the effect of inclusion or exclusion of additional modal degrees of freedom. Although not

of primary importance, this model also demonstrated aerodynamic nonlinearities.

4.1 Two Degree-of-Freedom Van der Pol Style Oscilla-

tor

In all the papers found about nonlinear normal modes to date, the only applications were in

structural dynamics with nonlinear stiffnesses. Before any more complex aeroelastic systems

were analyzed a simple system that would experience a limit cycle oscillation was sought.

The system chosen was a nonlinear, two degree-of-freedom, spring-mass-damper as seen in

Fig. 4.1. The left hand spring can be nonlinear by assigning a value to g. The damper is

made nolinear by a Van der Pol-like damping term. This is different from the standard Van

der Pol oscillator becuase it has two degrees of freedom so a normal mode can be defined.

The equations of motion for this system are

x1 = −(2k + gx1

2)x1 + kx2 −

(c+ µx1

2)x1

x2 = kx1 − 2kx2.(4.1)

In these equations, x1 and x2 are the degrees of freedom for the masses, k is the linear

spring stiffness, g is cubic nonlinear spring stiffness, c is the primary damping term, µ is the

nonlinear damping coefficient, and each overdot represents a time derivative.

4.2 Nonlinear Pitch-Plunge Airfoil with Quasi-Steady

Aerodynamics

The first aeroelastic model considered in this dissertation is a nonlinear variation of the

classic pitch-plunge airfoil as seen in Fig. 4.2. The non-dimensional equations of motion for

38


Figure 4.1: Nonlinear Spring-Mass-Damper

the pitch-plunge airfoil are

¨h+ xαα + ω2h = −L

xα¨h+ rα

2 α + rα2(1 +Gαα

2)α = M

(4.2)

These equations only differ from the classic equations that appear in Bisplinghoff et al. [42]

by the dimensionless, nonlinear stiffness term Gα. The variables in Eqs. (4.2) and Fig. 4.2

are as follows: h is the nondimensional plunge which is the plunge displacement h divided by

the half chord b, α is the pitch, Iα is the pitch inertia, m is the mass, xα is the dimensionless

static imbalance, Kα[α] = kα(1 + Gαα2), ω is the plunge-pitch natural frequency ratio ωh

ωα

where ωh =√

Khm

and ωα =√

kαIα

, rα is the dimensionless radius of gyration√

Iαmb2

, and an

overdot represents a time derivative. The nondimensional lift and moment appear as L and

M .

The aerodynamic lift and moment were modeled with Theodorsen’s lift and moment

39


Figure 4.2: Pitch-Plunge Airfoil Model

equations[43].

L =1

µ

(¨h+ uα− aα + 2uC[k]

(˙h+ uα +

(1

2− a)α

))M =

1

µ

(a¨h−

(1

2− a)uα−

(1

8+ a2

)α

)+ 2u

1

µ

(1

2+ a

)C[k]

(˙h+ uα +

(1

2− a)α

)(4.3)

In these equations u is the dimensionless freestream velocity, a is the elastic axis location,

C[k] is Theodorsen’s function, and µ is the density ratio. For quasi-steady aerodynamics,

C[k] is set to unity. The resulting quasi-steady aeroelastic system is directly compatible with

the NNM method and no further manipulation is required.

40


4.3 Nonlinear Pitch-Plunge Airfoil with Unsteady Aero-

dynamics

The pitch-plunge airfoil was also considered with unsteady aerodynamics. The same equa-

tions from the quasi-steady model, Eqs. (4.2) and (4.3), are used except C[k] is not set to

unity. For unsteady aerodynamics, C[k] is in the frequency domain which is not compatible

with the NNM method which requires analytic, time domain equations. To facilitate the

nonlinear normal mode method the lift and moment were rewritten as

L =1

µ

(¨h+ uα− aα + 2uLc

)M =

1

µ

(a¨h−

(1

2− a)uα−

(1

8+ a2

)α

)+ 2u

1

µ

(1

2+ a

)Lc

Lc = C[k]

(˙h+ uα +

(1

2− a)α

).

(4.4)

Venkatesan and Freidmann [44] developed approximate transfer functions representing Theodorsen’s

function through the use of Bode plots. Using two zeros and poles, Theodorsen’s function

can be written as

C[k] =0.5(ik + 0.135)(ik + 0.651)

(ik + 0.0965)(ik + 0.4555). (4.5)

The paper showed very good agreement between the transfer function and the exact Theodorsen’s

equations.

To model Lc, the transfer function was nondimensionalized and then converted to a

41


state space model with two states. The state space model can be written as x3

x4

= u

−0.0965 0.08676

0 −0.4555

x3

x4

+√u

0.09811

0.2211

( ˙h+ uα +

(1

2− a)α

)

Lc =√u[

0.1962 0.4422] x3

x4

+ 0.5

(˙h+ uα +

(1

2− a)α

).

(4.6)

The aerodynamic states start at x3 because x1 and x2 are reserved for the modal degrees of

freedom of the pitch-plunge airfoil. This state space aerodynamic model was then incorpo-

rated into the pitch-plunge airfoil system of two coupled, second-order differential equations

to create a system of four equations with two second-order and two first-order differential

equations. The full aeroelastic system is

x1 + xαx2 + ω2(1 +Ghx1

2)x1 = −L

xαx1 + rα2 x2 + rα

2(1 +Gαx2

2)x2 = M

x3 = u (−0.0965x3 + 0.08676x4) + 0.09811√u

(x1 + ux2 +

(1

2− a)x2

)x4 = −0.4555ux4 + 0.2211

√u

(x1 + ux2 +

(1

2− a)x2

),

(4.7)

where

L =1

µ

(x1 + ux2 − ax2 + 2u

(√u (0.1962x3 + 0.4422x4) + 0.5

(x1 + ux2 +

(1

2− a)x2

)))M =

1

µ

(ax1 −

(1

2− a)ux2 −

(1

8+ a2

)x2

)+ 2u

1

µ

(1

2+ a

)(√u (0.1962x3 + 0.4422x4) + 0.5

(x1 + ux2 +

(1

2− a)x2

)).

(4.8)

42


Figure 4.3: Goland Wing Model

4.4 Goland Beam Wing with Nonlinear, Quasi-Steady

Aerodynamics

The final model is like the Goland wing model [45]. The Goland wing model is a straight,

non-tapered, bending-torsion beam. In this case, the beam wing remains linear and the

nonlinearities appear in the aerodynamic model. A diagram of the beam wing model appears

in Fig. 4.3.

The equations of motion for the Goland beam wing are

m∂2w[y, t]

∂t2− Sy

∂2θ[y, t]

∂t2+ EI

∂4w[y, t]

∂y4= L

Iy∂2θ[y, t]

∂t2− Sy

∂2w[y, t]

∂t2−GJ ∂

2θ[y, t]

∂y2= M.

(4.9)

In these equations θ[y, t] is the pitch degree of freedom, w[y, t] is the plunge degree of freedom,

m is the mass per unit span, Iy is the moment of inertia in pitch, Sy is the static mass

imbalance, EI is the bending stiffness, GJ is the torsional stiffness, and L and M are the

lift and moment per unit span. Since the nonlinearities appear in the lift and moment terms

L and M, these equations are exactly as they appear in the linear model presented in [42].

For the beam wing, a quasi-steady aerodynamic model with a cubic stall-like nonlin-

43


earity was used.

L = ρV∞2bClα

(αe − C3αe

3)

M = −ρV∞2b2Cmα

(αe − C3αe

3)

αe = θ[y, t]− 1

V∞

∂w[y, t]

∂t

(4.10)

In these equations: ρ is air density, V∞ is the freestream velocity, αe is the effective angle

of attack , b is the semi-chord of the wing, Clα is the linear lift curve slope 2π, and Cmα is

the linear moment curve slope(12

+ a)Clα. A similar model was used by Kim and Strganac

[23], [24] and the value for the nonlinear aerodynamic term C3 = .00034189(180π

)3/Clα was

taken from their papers. The model uses strip theory which implies that two-dimensional

aerodynamics is sufficient to model the lift at each span location. Since strip theory does

not model wing tip vortices, the aspect ratio of the wing has to be relatively large for the

method to be reasonably accurate.

As shown above, the beam model obviously cannot be used with the nonlinear normal

modes method presented in the last chapter. The most convenient way to put the beam model

in a form where nonlinear normal modes can be applied is to use the assumed modes method

with Lagrange’s equations. For the assumed modes, a set of orthogonal polynomials that

satisfied all the beam’s boundary conditions were used. Up to five modes for bending and

five modes for torsion were used. The orthogonal polynomials were chosen over the exact

beam bending and torsion mode shapes because the polynomials process much faster. The

beam bending and torsion can then be written

w[y, t] =Nw∑i=1

φwi[y]ηi[t]

θ[y, t] =

Nθ∑i=1

φθi[y]η(i+Nw)[t]

(4.11)

where φwi[y] are the orthogonal polynomials for bending, φθi[y] are the orthogonal polynomi-

als for torsion, ηi[t] are the modal degrees of freedom, Nw is the number of bending modes,

44


and Nθ is the number of torsion modes.

The kinetic and potential energies of the beam are

T =1

2

l∫0

(∂w[y,t]∂t

∂θ[y,t]∂t

) m −Sy−Sy Iy

∂w[y,t]∂t

∂θ[y,t]∂t

dy (4.12)

and

U =1

2

l∫0

(∂2w[y,t]∂y2

∂θ[y,t]∂y

) EI 0

0 GJ

∂2w[y,t]∂y2

∂θ[y,t]∂y

dy (4.13)

and the generalized forces are

Qwi =

l∫0

φwi[y]Ldy, for i = 1, ..., Nw

Qθi =

l∫0

φθi[y]Mdy, for i = 1, ..., Nθ.

(4.14)

In these equations, l is the length of the beam and L and M are the lift and moment from

the nonlinear, quasi-steady aerodynamic model above. Lagrange’s equations of motion can

now be written.

∂

∂t

(∂(T − U)

∂ηi

)−(∂(T − U)

∂ηi

)= Qwi, for i = 1, ..., Nw

∂

∂t

(∂(T − U)

∂ηi

)−(∂(T − U)

∂ηi

)= Qθi, for i = 1 +Nw, ..., Nw +Nθ

(4.15)

Lagrange’s equations generate a system of Nw + Nθ nonlinear equations. The mass matrix

is linear so it can easily be extracted from the system and the equations can be rewritten as

[M ] {ηi} = {fi [{ηi} , {ηi}]} , for i = 1, ..., Nw +Nθ (4.16)

where [M ] is the mass matrix and fi are nonlinear functions of the modal positions and

45


velocities. This discretized version of the beam wing is now ready for the application of

nonlinear normal modes.

46

Chapter 5

Results

This chapter details the results from using the nonlinear normal mode (NNM) method to

predict limit cycle oscillation (LCO) in a series of different systems. Since nonlinear normal

modes have not previously been used for modeling of limit cycle oscillation, the first section

discusses a simple spring-mass-damper system with a Van der Pol type nonlinear damping

term. Linear and nonlinear system parameters were varied to study the effect on the accu-

racy of the NNM solution. The NNM method was also applied to two pitch-plunge airfoil

systems. Sample results are presented for both quasi-steady and unsteady aerodynamics.

The results from a study on the effect of changing the NNM master coordinate are included.

Improvements gained with Galerkin and collocation solutions for the modal coefficients are

also presented. The final section details the results from the NNM method used on a beam

wing and demonstrates that modes can be eliminated without a significant loss in accuracy

based on the modal coefficient solutions.

Several different solutions are compared in this chapter. All modal solutions are

compared to a reference solution which is a fourth-order Runge-Kutta time simulation of

each systems exact equations of motion (referred to as “Exact” in tables and plots) . The

Runge-Kutta solution was computed with both absolute and relative error goals set to half

of machine precision. The same Runge-Kutta scheme was used to simulate the all the modal

equations as well. Many of the nonlinear normal mode solutions are compared to a linear

47

Chapter 5. Results

normal mode solution. The linear normal modes are calculated the same way as the nonlinear

normal modes except that only the linear terms in the modal approximation function are re-

tained. In figures the linear normal mode solution is denoted as LNM. There are also several

different types of nonlinear normal mode solutions presented including: asymptotic nonlin-

ear normal modes (NNM-A), Galerkin nonlinear normal modes (NNM-G), and collocation

nonlinear normal modes (NNM-C).

5.1 Two Degree-of-Freedom

Van der Pol Style Oscillator

The primary goal of the two degree of freedom system was to show that the nonlinear normal

modes could model limit cycle oscillation and to explore possible limitations. Once the ability

of the nonlinear normal modes to predict LCO was established, the stiffness and damping

parameters and the initial conditions of the system were parametrically varied to see what

effect different relative values of linear to nonlinear terms would have on the accuracy of the

solution. The accuracy was judged by the amplitudes of the two masses and the frequency of

the motion once the final limit cycle oscillation was established. A small modal displacement

was used as the initial condition in all cases.

Table 5.1 contains the results from all the cases where LCO was identified. It contains

sections for parametric variation of the linear stiffness, the nonlinear stiffness, and linear and

nonlinear damping with and without the inclusion of a nonlinear stiffness term. Since the

system had two degrees of freedom, two nonlinear normal modes existed and results for both

are presented. All results use the first degree-of-freedom as the master coordinate.

Figures 5.1 through 5.8 present the results from two of the cases in the table. The

coefficients for these two cases are bolded. The results from these figures will be discussed

first followed by a discussion of the rest of the results in table 5.1.

Case 1 is an example of good results and can be see in phase space diagrams in Figs. 5.1

and 5.2 and time histories in Figs. 5.5 and 5.6. Case 1 used k = 10, g = 0.5, c = −0.3, and

48

Chapter 5. Results

µ = 0.6. The phase space diagrams show that there is good agreement between the NNM

and the exact simulations once the motion has settled into LCO. Both the amplitude and

the shape of the motion match closely. The time history plots further show that the NNM

captures the result well during the growth of the motion from a small initial condition.

Case 2 is an example of poor results and can be seen in phase space diagrams in

Figs. 5.3 and 5.4 and in time histories in Figs. 5.7 and 5.8. Case 2 used k = 10, g = 0.5,

c = −0.6, and µ = 0.3. In this case the phase diagrams show a very poor result on the part

of the NNM solution. The amplitudes are in error by a significant percentage and in the case

of the x2 the shape is totally different, especially in the first mode. The time histories show

that the NNM solution does a good job until the motion reaches a particular amplitude. This

indicates that the nonlinearities in the system have become too strong for the asymptotic

solution method. The error gets worse as the amplitude further increases since the strength

of the nonlinearity depends on the amplitude.

Variation of the initial condition is not presented in table 5.1 because it was found

to have no effect on the accuracy in either mode with one small exception. If the modal

displacement for the second mode was increased far enough the physical solution sometimes

ended up in the first mode. Since the nonlinear normal modes were approximate, the modal

initial conditions were composed of a mixture of the two modes when converted to the

physical coordinates and the physical system picked the mode it preferred.

Varying the stiffness and damping parameters did create a few problems not identified

in table 5.1. For certain ranges of values, the physical solution would experience what

appeared to be chaotic motion. There were other cases where the amplitude varied over

time which is indicative of internal resonances in the system. In either of theses cases, the

nonlinear normal modes are not the correct theoretical tool. In all the cases where the

chaotic motion was observed, the nonlinear normal mode exhibited divergent oscillations. In

cases with apparent internal resonances, the nonlinear normal mode produced a single an

LCO exhibiting a constant amplitude instead of the cyclically varying amplitude of the exact

solution.

49

Chapter 5. Results

With a close study of the data in table 5.1, a few trends can be found. In all the

cases were a stable LCO was found the frequency results were quite good even if the NNM

amplitude was in error by a substantial amount. The first variation shows an increase in the

linear stiffness with nonlinearity in damping only. As the linear stiffness increases the error

of the NNM solution decreases. The reduction in error can be explained by the reduction

in both the nonlinear damping coefficient and the destabilizing linear damping coefficient

(damping ratio has stiffness in the denominator). Increasing the nonlinear stiffness produces

a proportional increase in the NNM error. This is expected since the asymptotic method is

only valid for relatively weak nonlinearities. A nonlinear stiffness greater than ten percent

of the linear stiffness results in chaotic motion of this system which obviously cannot be

captured by a NNM solution. An increase in the linear damping of the system, making it

more unstable at the origin, increases the amplitude and degrades the results of the NNM

solution. This leads to the conclusion that a decrease in the linear stability of the system

adversely affects the accuracy of the asymptotic NNM solution. In effect, the nonlinear

damping term has to work harder to hold the system in LCO which creates a greater relative

nonlinearity in the system. An increase in the nonlinear damping term has no measurable

effect on the error when the stiffness is linear and actually reduces the error when the stiffness

is nonlinear. The increased nonlinear damping does decrease the amplitude of the motion in

both cases. In the nonlinear stiffness case, the improvement in results is likely due to the fact

that the amplitude is reduced resulting in less nonlinear stiffness effect. As seen before, the

nonlinear stiffness quickly increased the error so reducing its effect allows for an improvement

in results. Overall, it can be seen that the effect of changing the system parameters would

be quite difficult to predict a priori.

50

Chapter 5. Results

Table 5.1: NNM Results for SMD with Nonlinear Damping

k g c µ x1 Amp. x2 Amp. Freq. x1 Amp. x2 Amp. Freq. x1 Amp. x2 Amp. Freq.

1 0 -0.3 0.3 2.00 1.991 0.1579 1.844 1.937 0.1559 7.89 2.70 1.265

5 0 -0.3 0.3 2.00 1.998 0.356 1.963 1.986 0.352 1.855 0.612 1.081

10 0 -0.3 0.3 2.00 1.999 0.503 1.981 1.993 0.501 0.950 0.312 0.516

10 0 -0.3 0.3 2.00 1.999 0.503 1.981 1.993 0.501 0.950 0.312 0.516

10 0.5 -0.3 0.3 1.987 2.12 0.521 1.869 1.988 0.518 5.91 6.25 0.680

10 1 -0.3 0.3 1.975 2.25 0.536 1.762 1.953 0.528 10.78 13.09 1.405

1 0 -0.3 0.3 2.00 1.991 0.1579 1.844 1.937 0.1559 7.89 2.70 1.265

1 0 -0.6 0.3 2.84 2.78 0.1581 2.24 2.54 0.1505 21.1 8.65 4.83

1 0 -0.3 0.6 1.42 1.408 0.1579 1.304 1.370 0.1558 7.89 2.70 1.328

1 0 -0.6 0.6 2.01 1.967 0.1581 1.583 1.796 0.1510 21.1 8.65 4.46

10 0.5 -0.3 0.3 1.987 2.12 0.521 1.869 1.988 0.518 5.91 6.25 0.680

10 0.5 -0.6 0.3 2.79 3.17 0.535 2.44 2.73 0.525 12.55 13.89 1.885

10 0.5 -0.3 0.6 1.409 1.456 0.512 1.361 1.411 0.510 3.41 3.10 0.478

10 0.5 -0.6 0.6 1.987 2.12 0.520 1.827 1.965 0.512 8.09 7.19 1.413

k g c µ x1 Amp. x2 Amp. Freq. x1 Amp. x2 Amp. Freq. x1 Amp. x2 Amp. Freq.

1 0 -0.3 0.3 2.00 2.01 0.276 1.999 2.20 0.280 0.065 9.18 1.211

5 0 -0.3 0.3 2.00 2.00 0.616 2.000 2.04 0.618 0.023 1.84 0.316

10 0 -0.3 0.3 2.00 2.00 0.872 2.000 2.02 0.873 0.012 0.922 0.172

10 0 -0.3 0.3 2.00 2.00 0.872 2.000 2.02 0.873 0.012 0.922 0.172

10 0.5 -0.3 0.3 1.996 1.850 0.883 2.19 2.02 0.886 9.49 9.01 0.317

10 1 -0.3 0.3 1.992 1.713 0.895 2.46 1.953 0.905 23.5 14.0 1.161

1 0 -0.3 0.3 2.00 2.01 0.276 1.999 2.20 0.280 0.065 9.18 1.211

1 0 -0.6 0.3 2.83 2.89 0.275 2.85 4.01 0.288 0.776 38.7 4.66

1 0 -0.3 0.6 1.414 1.422 0.276 1.413 1.553 0.280 0.066 9.18 1.211

1 0 -0.6 0.6 2.00 2.04 0.275 2.02 2.83 0.288 0.777 38.7 4.56

10 0.5 -0.3 0.3 1.996 1.850 0.883 2.19 2.02 0.886 9.49 9.01 0.317

10 0.5 -0.6 0.3 2.82 2.43 0.894 3.50 3.04 0.910 24.2 25.3 1.765

10 0.5 -0.3 0.6 1.413 1.360 0.877 1.474 1.430 0.879 4.37 5.13 0.245

10 0.5 -0.6 0.6 1.996 1.853 0.882 2.19 2.12 0.890 9.65 14.16 0.815

Damping with

Nonlinear

Stiffness

Variation

Linear Stiffness

Nonlinear

Stiffness

Damping with

Linear Stiffness

Percent ErrorNNM-AExact Equations

Percent Error

Mode 1

Mode 2

NNM-AExact Equations

Damping with

Nonlinear

Stiffness

Variation

Linear Stiffness

Nonlinear

Stiffness

Damping with

Linear Stiffness

51

Chapter 5. Results

-1.0 -0.5 0.5 1.0x1

-4

-2

2

4

x 1

Exact

NNM-A

(a) x1 Phase Space

-1.5 -1.0 -0.5 0.5 1.0 1.5x2

-4

-2

2

4

x 2

(b) x2 Phase Space

Figure 5.1: Nonlinear SMD Phase Space Diagrams for First NNM, Case 1

52

Chapter 5. Results

-1.5 -1.0 -0.5 0.5 1.0 1.5x1

-5

5

x 1

Exact

NNM-A

(a) x1 Phase Space

-1.0 -0.5 0.5 1.0x2

-5

5

x 2

(b) x2 Phase Space

Figure 5.2: Nonlinear SMD Phase Space Diagrams for Second NNM, Case 1

53

Chapter 5. Results

-2 -1 1 2x1

-5

5

x 1

Exact

NNM-A

(a) x1 Phase Space

-3 -2 -1 1 2 3x2

-10

-5

5

10

x 2

(b) x2 Phase Space

Figure 5.3: Nonlinear SMD Phase Space Diagrams for First NNM, Case 2

54

Chapter 5. Results

-3 -2 -1 1 2 3x1

-20

-10

10

20

x 1

Exact

NNM-A

(a) x1 Phase Space

-3 -2 -1 1 2 3x2

-15

-10

-5

5

10

15

x 2

(b) x2 Phase Space

Figure 5.4: Nonlinear SMD Phase Space Diagrams for Second NNM, Case 2

55

Chapter 5. Results

10 20 30 40 50 60 70t

-1.0

-0.5

0.5

1.0

x1

Exact

NNM-A

(a) x1 Time History

10 20 30 40 50 60 70t

-1.5

-1.0

-0.5

0.5

1.0

1.5x2

(b) x2 Time History

Figure 5.5: Nonlinear SMD Time History for First NNM, Case 1

56

Chapter 5. Results

10 20 30 40 50 60t

-1.5

-1.0

-0.5

0.5

1.0

1.5x1

Exact

NNM-A

(a) x1 Time History

10 20 30 40 50 60t

-1.0

-0.5

0.5

1.0

x2

(b) x2 Time History

Figure 5.6: Nonlinear SMD Time History for Second NNM, Case 1

57

Chapter 5. Results

10 20 30 40 50t

-2

-1

1

2

x1

Exact

NNM-A

(a) x1 Time History

10 20 30 40 50t

-3

-2

-1

1

2

3

x2

(b) x2 Time History

Figure 5.7: Nonlinear SMD Time History for First NNM, Case 2

58

Chapter 5. Results

10 20 30 40 50t

-3

-2

-1

1

2

3

x1

Exact

NNM-A

(a) x1 Time History

10 20 30 40 50t

-3

-2

-1

1

2

3

x2

(b) x2 Time History

Figure 5.8: Nonlinear SMD Time History for Second NNM, Case 2

59

Chapter 5. Results


with Quasi-steady Aerodynamics

The pitch-plunge airfoil with quasi-steady aerodynamics an a nonlinear pitch stiffness was

the first aeroelastic system on which the NNM method was attempted. Since it was the

simplest aeroelastic system, it was also used for studies on master coordinate selection and

improved coefficient solution methods.

For the test cases presented, the following constants were used in Eq. (4.2).

µ = 11

a = −0.35

xα = .2

rα = .5

ω = 0.5

Gα = 0.5

(5.1)

Additionally C[k] was set to unity making the aerodynamics quasi-steady. This model has a

dimensionless linear flutter speed of 0.807 and linear flutter frequency of 0.1598 Hz. Other

cases were tested, but the results were very nearly the same and only confirmed the conclu-

sions reached with the model above.

5.2.1 Sample Results

Since the system in Eq. (4.2) is not in the form needed for the NNM method, it must first

be transformed. The simplest route is to multiply by the inverse of the mass matrix so the

plunge degree of freedom can be used as the master coordinate. At each freestream velocity

of interest, the standard derivation of the asymptotic NNMs was followed. The final result

is two uncoupled modal equations. Below is an example of this result for a nondimensional

60

Chapter 5. Results

velocity of 1.05 times the linear flutter velocity. The LCO NNM modal equation is

u =u(−1.000− 4.37u2 − 9.44u2 − 26.5u4 − 62.7u2u2 − 7.45u4

− 192.1u6 − 272u4u2 + 155.2u2u4 + 192.8u6

− 465u8 − 1288u6u2 − 1373u4u4 − 900u2u6 − 331u8)

+ u(0.00462− 2.86u2 + 0.312u2 − 68.5u4 − 44.3u2u2 + 25.5u4

− 206u6 − 231u4u2 − 76.0u2u4 − 69.3u6

+ 207u8 + 1091u6u2 + 1475u4u4 + 636u2u6 + 62.9u8).

(5.2)

The destabilizing linear damping term is apparent in this LCO modal equation. The second

NNM modal equation is a damped oscillator as would be expected.

u =u(−0.236− 0.01472u2 − 0.0316u4 + 0.01460u6 − 0.00224u8

)+ u

(−0.207 + 0.00738u2 − 0.00224u4 + 0.0000965u6 − 1.347× 10−6u8

)+ (0.00179u3 − 0.000434u5 + 0.00001555u7 − 1.855× 10−7u9 − 0.00874u3u2

+ 0.000555u5u2 − 9.72× 10−6u7u2 − 0.00469u3 − 0.0239u2u3 + 0.00207u4u3

− 0.0000443u6u3 + 0.00549u3u4 − 0.0001631u5u4 − 0.01516u5 + 0.01151u2u5

− 0.000483u4u5 − 0.001066u3u6 + 0.00738u7 − 0.001865u2u7 − 0.001197u9)

(5.3)

The stabilizing linear damping is easily visible in this damped modal equation. For compar-

ison, the modal equations of motion for the LNM are also presented. First, the LCO mode

is

u = u(−1.000− 1.215u2 − 6.83u2

)+ u

(0.00462− 4.99u2 − 3.12u2

). (5.4)

The damped modal equation for LNM is

u = u(−0.236 + 0.0228u2

)+ u

(−0.207 + 0.01661u2

)+(0.00404u3 + 0.01039u3

)(5.5)

61

Chapter 5. Results

The stabilizing and destabilizing linear damping terms can also be seen in these LNM equa-

tions. Comparing these two sets of equations, it is obvious that the NNM equations capture

significantly more nonlinear coupling terms than the LNM equations.

The Runge-Kutta simulation of the NNM modal equation for LCO is compared to a

simulation of the original equation of motion and the LNM equation of motion for LCO. The

exact solution is given a modal initial condition for the LCO NNM. Figure 5.9 shows the

growth of LCO motion and 5.10 shows in phase space the motion after settling into LCO.

As can be seen in the time history and phase space plots, the NNM does a very good job of

modeling LCO motion and growth. The NNM is obviously much better than the LNM. The

reason the LNM equation does not do well is obvious when Eqs. (5.2) and (5.4) are compared

since the LNM equation is missing a vast majority of the nonlinear terms.

Figure 5.11 shows a comparison of the time history for the damped modes. Both the

NNM and the LNM give a nearly exact solution for the damped mode’s decay. All of this

motion occurs in the linear amplitude range and since the linear parts of both solutions are

exact, a nearly exact solution is produced. These are not exact solutions since the nonlinear

terms still exist and are inexact even though they are very, very small. Due to the inexactness

of the NNM, which is used to generate the modal initial condition for the simulation of the

original equations of motion, an extremely small part of the LCO mode is excited by the

initial condition. If the simulation of the exact equations of motion is allowed to run for long

enough, the LCO motion will begin to appear. This is shown in Fig. 5.12. Since this growth

is due to inaccuracies in the initial condition, the modal solutions do not show this motion

growth.

The results for the NNM are not always as good as in the above example. If the

nonlinearities in the system are too strong, the asymptotic NNM will not perform as well.

Figure 5.13 is the same system as above except the nondimensional freestream velocity has

been increased to 1.17 times the linear flutter velocity. The higher velocity causes a higher

amplitude. This makes the system nonlinearities stronger since the nonlinearity is dependent

on the amplitude. In the figure, it is obvious that neither the NNM or LNM are doing a

62

Chapter 5. Results

50 100 150 200t

-0.06

-0.04

-0.02

0.02

0.04

0.06

h

Exact

LNM

NNM-A

(a) Plunge

50 100 150 200t

-0.2

-0.1

0.1

0.2

Α

(b) Pitch

Figure 5.9: LCO Time History Plots of Quasi-steady Airfoil at 1.05 Vf

63

Chapter 5. Results

-0.10 -0.05 0.05 0.10h

-0.10

-0.05

0.05

0.10

h

Exact

LNM

NNM-A

(a) Plunge

-0.3 -0.2 -0.1 0.1 0.2 0.3 0.4Α

-0.4

-0.2

0.2

0.4

Α

(b) Pitch

Figure 5.10: LCO Phase Space Plots of Quasi-steady Airfoil at 1.05 Vf

64

Chapter 5. Results

10 20 30 40 50t

-0.10

-0.05

0.05

0.10

0.15

0.20

h

Exact

LNM

NNM-A

(a) Plunge

10 20 30 40 50t

-0.04

-0.02

0.02

0.04

0.06

Α

(b) Pitch

Figure 5.11: Damped Mode Time History Plots of Quasi-steady Airfoil at 1.05 Vf

65

Chapter 5. Results

2000 4000 6000 8000t

-0.10

-0.05

0.05

0.10

h

Figure 5.12: Growth of LCO Motion in Exact System due to Approximate Initial Condition

good job of modeling the exact solution, but the NNM is doing much better than the LNM.

Figures 5.14 and 5.15 show the exact and asymptotic NNM frequencies and amplitudes as

the freestream velocity is increased. These figures also display the degradation of the NNM

solution when the velocity is increased. If the velocity is increased further, the asymptotic

solution will slowly lose its stability and produce a divergent solution even though a LCO

still exists for the exact solution.

Figures 5.16 and 5.17 show the convergence of the solution as the order of the ap-

proximation is increased. At 1.01Vf , the solution quickly converges toward zero as the order

is increase. However, the solution does not immediately converge at 1.05Vf . The error first

increases and then converges toward zero.

66

Chapter 5. Results

-0.15 -0.10 -0.05 0.05 0.10 0.15h

-0.2

-0.1

0.1

0.2

h

Exact

LNM

NNM-A

(a) Plunge

-0.6 -0.4 -0.2 0.2 0.4 0.6Α

-0.6

-0.4

-0.2

0.2

0.4

0.6

Α

(b) Pitch

Figure 5.13: LCO Phase Space Plots of Quasi-steady Airfoil at 1.17 Vf

67

Chapter 5. Results

0.16

0.162

0.164

0.166

0.168

0.17

0.172

0.174

0.176

0.178

1 1.05 1.1 1.15 1.2

V/Vf

Frequency (Hz)

Exact

NNM-A

Figure 5.14: Frequency vs. Velocity

0

0.05

0.1

0.15

0.2

0.25

1 1.05 1.1 1.15 1.2

V/Vf

Nondimensional Amplitude

Exact

NNM-A

Figure 5.15: Amplitude vs. Velocity

68

Chapter 5. Results

0

0.05

0.1

0.15

0.2

0.25

3 5 7 9

Order of Approximation

Percent Error

Frequency

Position Amp.

Velocity Amp.

Figure 5.16: Asymptotic Convergece at 1.01Vf

0

0.2

0.4

0.6

0.8

1

1.2

3 5 7 9

Order of Approximation

Percent Error

Frequency

Position Amp.

Velocity Amp.

Figure 5.17: Asymptotic Convergece at 1.05Vf

69

Chapter 5. Results

5.2.2 Master Coordinate Study

Each nonlinear normal mode is built off a specific master coordinate. Since the resulting mode

is an approximation, the selection of the master coordinate affects the accuracy of the results.

The master coordinate can be directly a degree of freedom or it can be a modal coordinate.

Throughout this work three different modal coordinates were used for the pitch-plunge airfoil

model. They are referred to as M−1, linear structural mode shapes with apparent mass, and

linear flutter mode. Each is discussed in detail below.

Direct Degree-of-Freedom (M−1)

The easiest master coordinate to use is one of the degrees of freedom of the system. However,

unlike in the spring-mass-damper example, the mass matrix for an aeroelastic system is

usually not an identity matrix. The aeroelastic system from Eq. (4.2) was rewritten as

[M ] {qi} = fi [{qi} , {qi}] . (5.6)

Note that the mass matrix [M ] above is the aeroelastic mass matrix and thus includes the

apparent mass effects from the aerodynamic model. Both sides of the rewritten equation are

then multiplied by the inverse of the mass matrix to get

{qi} = [M ]−1fi [{qi} , {qi}] . (5.7)

Now one of the degrees of freedom may be used as the master coordinate. This master

coordinate may be referred to as the M−1, plunge, or pitch master coordinate.

Linear Structural Mode Shapes with Apparent Mass

The second master coordinate used was referred to as the linear structural mode shapes with

apparent mass effects. This is a modal degree of freedom used as the master coordinate.

To find this modal coordinate, the aeroelastic system in Eq. (4.2) was linearized and the

70

Chapter 5. Results

freestream velocity was set to zero. 1 + 1µ

xα − aµ

xα − aµ

rα2 + 1

8µ+ a2

µ

h

α

=

0 0

0 0

h

α

− ω2 0

0 rα2

h

α

(5.8)

A standard linear modal analysis was preformed and the modal transformation matrix was

found. The linear modal transformation matrix was used to transform the nonlinear system

which is in the form of Eq. (5.6). Since the aeroelastic mass matrix is linear and does not

depend on the velocity, the transformation converts it to an identity matrix and system will

be of the nice form below.

{ηi} = fi [{ηi} , {ηi}] (5.9)

With the nonlinear system written in terms of the linear structural modes with apparent

mass, ηi, each of the modal coordinates could be used as the master coordinate.

Linear Flutter Mode

The linear flutter mode was also used as a modal master coordinate. Starting with the

aeroelastic equations of motion in Eq. (4.2), the system was linearized. The velocity of

interest (above the linear flutter velocity) was chosen and the eigenvalue procedure detailed

in Patil[46] was used to find the left and right modal transformation matrices required to put

the linear system in decoupled, second-order form. When these transformation matrices are

applied to a linearized system in first-order form, it results in a block diagonal matrix with

sub-matrices of the form: 0 1

C1 C2

. (5.10)

When these same transformation matrices are applied to the nonlinear first-order system

matrix, the resulting matrix is fully populated. Assuming a cubic pitch stiffness nonlinearity,

71

Chapter 5. Results

the block diagonal parts take the form seen below and the off block diagonal zeros are

constants times α2. C3α2 1 + C4α

2

C1 + C5α2 C2 + C6α

2

(5.11)

This first-order system of equations obviously cannot be converted back to second-order form

and is consequently not in the form needed for the standard derivation of NNM. Thus, the

first-order formulation had to be utilized anytime the linear flutter mode was used as the

master coordinate. When choosing the two master coordinates from the first-order system,

the two equations that would create the flutter mode from the linear matrix must be selected.

If the other pair that creates the damped mode is selected, the LCO will not be found.

Additionally, the linear unknown modal coefficients will be exactly zero since there is no

linear coupling between the modes.

Master Coordinate Results

Figures 5.18 and 5.19 show the results from the master coordinate study. Note that the

vertical axis is logarithmic. The percent error of the NNM solution is plotted against in-

creasing velocity. The errors are found by comparing the NNM solution to the simulation

of the original equations of motion. The percent error was calculated for the frequency, the

displacement amplitude of the master and slave coordinates, and the velocity amplitude of

the master and slave coordinates. The average of all four amplitude errors gave an accurate

picture of how the NNM was performing with out have to look at a whole handful of plots;

therefore, the average amplitude error is presented in the plot. The master coordinates are

as follows, Minv is the plunge, Minv 2 is the pitch, ZV and ZV 2 are the first and second

linear structural modes with apparent mass, and flutter is the linear flutter mode.

It is obvious from the figures that the accuracy of the results is dependent on the

selection of the modal coordinate. It is not immediately apparent what the best coordinate

would be. Selection would appear to depend on what velocity was of interest. The results

72

Chapter 5. Results

Asymptotic Master Coordinate Study

0.01

0.10

1.00

10.00

100.00

0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19

U/Vf

Amplitude Percent Error

Minv

Minv 2

ZV

ZV 2

Flutter

Figure 5.18: Master Coordinate Amplitude Error Results

for all of the master coordinates degrade as the velocity increases. This is caused by the

larger amplitude at the higher velocities creating a system that effectively has a stronger

nonlinearity. Within about 10% of the flutter velocity, the linear flutter mode gives the best

results for both amplitude and frequency. After this point the, linear flutter master coordinate

starts becoming unstable and produces a divergent solution at 1.19Vf (just beyond the range

of the plot). At a certain velocity (or amplitude), all of the NNMs based on an asymptotic

formulation will begin to diverge. In this case, the linear flutter master coordinate is the first

to lose it’s stability. The structural mode shapes are not far behind and produce a divergent

solution at 1.29Vf . The pitch and plunge master coordinates remain stable much longer

(> 2Vf ), but as can be seen in the error plots they have a large error even at 1.17Vf . Since

the flutter mode is the most accurate in the low velocity range where the asymptotic method

can be trusted, it is the best master coordinate for the low speed regime. Additionally, as

will be shown in the section on the collocation solution, the linear flutter mode exhibits an

apparent convergence and very low error when used with the collocation method while the

other mode shapes do not. With this information, the linear flutter mode can be considered

the best choice of master coordinate.

73

Chapter 5. Results

Asymptotic Master Coordinate Study

0.0001

0.001

0.01

0.1

1

10

0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19

U/Vf

Frequency Percent Error

Minv

Minv 2

ZV

ZV 2

Flutter

Figure 5.19: Master Coordinate Frequency Error Results

5.2.3 Galerkin Modal Coefficient Solution

The Galerkin solution method was used in an attempt to improve on the results of the

asymptotic method. Use of the Galerkin method should allow stronger nonlinearities in the

system without a large loss in accuracy. The Galerkin solution also remains stable at high

velocities. The Galerkin method can offer very good results, but has two main drawbacks.

The computational cost for the Galerkin method is quite high with the Galerkin solution

taking near ten minutes while the asymptotic solution can be found in about 0.1 seconds.

The Galerkin solution was also sensitive to the region used for integration and there is no

immediately apparent method to iterate the integration region and establish a convergence

criterion.

Galerkin Solution at Moderate Velocity

Table 5.2 contains the results from the Galerkin method used for the same airfoil system used

in the sample results section above. The moderate velocity, 1.17Vf , was used. The “Region -

Exact Ratio” column is the integration limits divided by the amplitude of the exact solution.

This gives the size of the integration region relative to the size of the exact LCO motion. The

74

Chapter 5. Results

Table 5.2: Galerkin Integration Region Data

U V u v u v u v u v

0.5 0.0913 0.1006 0.1579 0.1694 -13.53 -15.80 1.729 1.684

0.75 0.1369 0.1509 0.1361 0.1475 -25.45 -26.66 0.994 0.978

0.9 0.1643 0.1811 0.1454 0.1409 -20.36 -29.95 0.885 0.778

1 0.1826 0.201 0.1440 0.1522 -21.12 -24.36 0.789 0.756

1.1 0.201 0.221 0.1707 0.1886 -6.51 -6.27 0.850 0.852

1.2 0.219 0.241 0.1733 0.1811 -5.07 -9.99 0.791 0.750

1.3 0.237 0.262 0.1833 0.204 0.38 1.57 0.772 0.781

1.4 0.256 0.282 0.1832 0.203 0.36 1.03 0.717 0.722

1.5 0.274 0.302 0.1858 0.202 1.74 0.54 0.678 0.670

1.6 0.292 0.322 0.157695 0.1654 -13.63 -17.76 0.540 0.514

Solution-Region

FractionPercent Error

0.1826 0.201

Exact Solution

Amplitude

NNM-G Solution

AmplitudeIntegration LimitsRegion -

Exact Ratio

actual limits used for the Galerkin integrals are given in the next two columns. The percent

error for u and v is presented. Since the M−1 plunge master coordinate is being used, u

and v correspond directly to h and h. The last two columns in the table, “Solution - Region

Fraction,” is the Galerkin solution amplitudes divided by the integration limits. This number

shows how large the approximate LCO solution is compared to the integration region. As can

be seen by looking at these results, the error fluctuates as the size of the integration region

is increased. The solution-region fraction does not have any special relationship to the error.

The best results are found when the fraction is around 0.72, but a few rows up a fraction

of about 0.76 generates some of the worst results. Without a special relation between what

would be the only two known quantities in a problem where the exact solution is not known,

an iteration scheme and convergence criteria are difficult to create. Five of the solutions

from the table are plotted in Figs. 5.20 through 5.24. The rows corresponding to the plotted

solutions are in bold. The figures show that that the Galerkin solution can give very good

results, but if the integration region is not the correct size the Galerkin solution offers poor

results that no better than the asymptotic solution.

75

Chapter 5. Results

-0.15 -0.10 -0.05 0.05 0.10 0.15h

-0.2

-0.1

0.1

0.2

h

Exact

NNM-G

NNM-A

-0.6 -0.4 -0.2 0.2 0.4 0.6Α

-0.6

-0.4

-0.2

0.2

0.4

0.6

Α

Figure 5.20: Galerkin with Integration Region 0.5×Exact Amplitudes, 1.17Vf

76

Chapter 5. Results

-0.15 -0.10 -0.05 0.05 0.10 0.15h

-0.2

-0.1

0.1

0.2

h

Exact

NNM-G

NNM-A

-0.6 -0.4 -0.2 0.2 0.4 0.6Α

-0.6

-0.4

-0.2

0.2

0.4

0.6

Α


77

Chapter 5. Results

-0.15 -0.10 -0.05 0.05 0.10 0.15h

-0.2

-0.1

0.1

0.2

h

Exact

NNM-G

NNM-A

-0.6 -0.4 -0.2 0.2 0.4 0.6Α

-0.6

-0.4

-0.2

0.2

0.4

0.6

Α


78

Chapter 5. Results

-0.15 -0.10 -0.05 0.05 0.10 0.15h

-0.2

-0.1

0.1

0.2

h

Exact

NNM-G

NNM-A

-0.6 -0.4 -0.2 0.2 0.4 0.6Α

-0.5

0.5

Α


79

Chapter 5. Results

-0.15 -0.10 -0.05 0.05 0.10 0.15h

-0.2

-0.1

0.1

0.2

h

Exact

NNM-G

NNM-A

-0.6 -0.4 -0.2 0.2 0.4 0.6Α

-0.6

-0.4

-0.2

0.2

0.4

0.6

Α


80

Chapter 5. Results

Galerkin Solution at High Velocity

The velocity for the airfoil model was increased to 1.5Vf to determine the accuracy of the

Galerkin solution with even stronger nonlinearities. The integration region was varied over

the same range as in Table 5.2, but sastisfactory results were not found. Figure 5.25 shows

the best results that were obtained with the Galerkin method for this case. As is obvious

in the figure, the Galerkin solution is no better than the asymptotic at a high velocity. The

Galerkin integrals have to be taken over such a large region to encompass the LCO amplitude

that the actual error for the LCO ends up being quite high.

5.2.4 Collocation Modal Coefficient Solution

The collocation method was used in an attempt to improve the results from the asymptotic

method while avoiding the problems of the Galerkin method. Several results are presented

here that show how well the collocation method works. The same airfoil model as in the

previous sections was used.

Collocation Method at Moderate Velocities

To generate the collocation solution, a set of points must be chosen. At the moderate

velocity of 1.17 times the flutter velocity, the asymptotic solution can offer a starting point

for selecting these points. Even though the asymptotic solution has a fairly large error in this

regime, the information is good enough for a first collocation solution. Successive iterations

can be done by selecting the points from a collocation solution and repeating the collocation

method with the new set of points.

Table 5.3 shows the resulting error in both amplitude and frequency of LCO after

one, two, and three iterations of the collocation method. The amplitude error shown is an

average of the displacement and velocity amplitudes as it seemed to be the best indicator

of the solutions accuracy. Results are shown with both the linear flutter mode and airfoil

plunge used as the master coordinate. The plunge master coordinate data can be compared

81

Chapter 5. Results

-0.3 -0.2 -0.1 0.1 0.2 0.3h

-0.4

-0.2

0.2

0.4

h

Exact

NNM-G

NNM-A

-1.0 -0.5 0.5 1.0Α

-1.0

-0.5

0.5

1.0

1.5Α


82

Chapter 5. Results

Table 5.3: Collocation Results

Frequency Amplitude

Asymptotic 1.758 11.035

Collocation 1 0.135 0.794



Asymptotic 3.760 22.150




Plunge

Master

Coordinate

SolutionPercent Error

Flutter

Master

Coordinate

directly to the previous sections since it is the exact same case and master coordinate. For

each row in the table, an equally spaced set of points for the collocation solution was taken

from two adjacent quadrants in the previous LCO solution. The first collocation iteration

shows an excellent result in both master coordinates. With the plunge master coordinate,

the error may increase or decreases for additional iterations, but generally trends downward.

For the flutter mode master coordinate, additional collocation iterations provide a further

decrease in error. Convergence for this technique has not been proven, but existed for several

other cases tested.

Figure 5.26 compares the steady state LCO of the collocation solution using the linear

flutter mode master coordinate to the asymptotic and exact results at a freestream velocity

of 1.17 times the flutter velocity. This figure corresponds to the first flutter mode, colloca-

tion iteration in table 5.3. A single iteration of the collocation method shows dramatically

improved results. The amplitude in this case is less than one percent in error.

83

Chapter 5. Results

-0.5 0.5Η1

-0.5

0.5

Η1

Exact

NNM-C

NNM-A

Figure 5.26: Collocation Modal Coefficient Solution Sample Results at 1.17Vf

84

Chapter 5. Results

Collocation Method and Motion Growth

Since the collocation method was meant to push the error to zero on the LCO path, the

accuracy of growth of the motion from the initial condition was not assured. However, the

growth was actually modeled very accurately. Using a single collocation iteration, Figs.

5.27(a) through 5.27(c) were generated to show the accuracy in the growth phase of the

motion from a small initial condition. After the third growth figure, the motion quickly

stabilizes into the LCO shown in Fig. 5.26. As the figures show, the error during the growth

is very small. The error in the moderate amplitudes has no chance to grow because it is

sandwiched between two areas where the error is forced towards zero, the linear range due

to exact linear coefficients and the band where the LCO occurs due to the selection of the

collocation points. This demonstrates that the collocation method works very well for the

growth phase as well as steady LCO.

Collocation Method at High Velocities

In addition to the reduction in error at low velocities where the asymptotic solution is stable,

the collocation solution remains stable at high amplitudes and velocities where the asymptotic

solution is no longer stable and the Galerkin solution has not been found to offer satisfactory

results. With the appropriate selection of ui and vi points, the collocation method offers a

solution of very high accuracy even at very high velocities. However, no direct way exists to

select points for the first iteration of the collocation solution. If the points chosen are too

much smaller in magnitude than the LCO, the collocation solution will be unstable just like

the asymptotic. Nonetheless, with a little trial and error and a simple iteration scheme an

excellent solution can be found.

This example uses the same system as above, except the velocity has been increased to

1.5 times the linear flutter velocity. The linear flutter mode is used as the master coordinate.

The exact solution could be used to select the points needed for collocation, but this does

not offer a realistic example. So as a rough first guess, the collocation points from the 1.17

times the linear flutter example were doubled in magnitude. These points created a stable

85

Chapter 5. Results

20 40 60 80 100t

-0.2

-0.1

0.1

0.2

Η1

Exact

NNM-C

(a) 0 to 100 Seconds

120 140 160 180 200t

-0.4

-0.2

0.2

0.4

Η1

Exact

NNM-C

(b) 100 to 200 Seconds

220 240 260 280 300t

-0.6

-0.4

-0.2

0.2

0.4

0.6

Η1

Exact

NNM-C

(c) 200 to 300 Seconds

Figure 5.27: Motion Growth with Collocation Solution

86

Chapter 5. Results

collocation solution, seen in Fig. 5.28, although the error was relatively high at about 7%, 12%

and 26% for the frequency, displacement amplitude, and velocity amplitudes, respectively. If

a stable solution had not been found the magnitudes could have been increased further until

a stable solution was reached. The former iteration method of using the previous collocation

solution’s points to select the next set does not work in this case. Selecting these points

will generate an unstable solution. Instead, the points extracted from the new collocation

solution are averaged with the points from the previous solution. If a stable solution is

found, then this iteration continues. Sometimes an unstable solution will be generated. If

this occurs, the current unstable set of points is averaged with the previous stable set of

points. The iteration continues in this fashion until the percent step size for all the ui and

vi points is less than 10%. Table 5.4 demonstrates this process. The boxes containing only

dashes indicate that the point set was not extracted from the solution, but was taken from

the previous stable set of points. As can be seen the solution converges to a very accurate

result. Figures 5.28 through 5.30 correspond to the iterations listed in the table. A figure for

iteration 3 is not included since it produces an unstable solution. The figures, especially the

converged Fig. 5.30, clearly show the excellent accuracy of the collocation solution even for

the highly nonlinear solution that cannot even be approximately captured by the asymptotic

or Galerkin solutions.


with Unsteady Aerodynamics Sample Results

The constants from Eq. (5.1) were used with the unsteady aeroelastic equations. The dimen-

sionless linear flutter velocity for this system is 1.699 and the linear flutter velocity is 0.1387

Hz. Since the linear flutter mode has been established as the preferred master coordinate

and the first-order formulation already had to be used due to the unsteady aerodynamics,

the linear flutter mode was selected as the master coordinate for this test case. As previously

discussed in the master coordinate section, the two first-order modal degrees of freedom that

87

Chapter 5. Results

Table 5.4: Collocation Iteration at a High Velocity

ui vi ui vi ui vi

1.66 0 1.22 0

1.12 1.24 0.79 0.93

0 1.6 0 1.06

-1 1.16 -0.55 0.82

1.44 0 1.35 0 -13.3 -

0.96 1.09 0.87 1.04 -14.7 -12.5

0 1.33 0 1.17 - -16.9

-0.78 0.99 -0.56 0.92 -22.5 -14.7

1.40 0 - - -3.1 -

0.91 1.06 - - -4.5 -2.1

0 1.25 - - - -6.0

-0.67 0.96 - - -13.9 -3.5

1.42 0 1.45 0 -1.6 -

0.93 1.07 0.94 1.13 -2.2 -1.0

0 1.29 0 1.25 - -3.0

-0.72 0.97 -0.57 0.99 -6.9 -1.8

4 0.29 0.95 0.94

3unstable

solution

unstable

solution

unstable

solution

N/A N/A

2 3.082 6.485 10.888

1 6.689 16.297 25.598

Iteration

Collocatoin Points ErrorPercent Step Size

Used ExtractedFrequency

Displacement

Amplitude

Velocity

Amplitude

88

Chapter 5. Results

-1.5 -1.0 -0.5 0.5 1.0 1.5Η1

-1.5

-1.0

-0.5

0.5

1.0

1.5

Η1

Exact

NNM-C

Figure 5.28: Collocation Iteration 1 at 1.5Vf

89

Chapter 5. Results

-1.5 -1.0 -0.5 0.5 1.0 1.5Η1

-1.5

-1.0

-0.5

0.5

1.0

1.5

Η1

Exact

NNM-C


90

Chapter 5. Results

-1.5 -1.0 -0.5 0.5 1.0 1.5Η1

-2

-1

1

2

Η1

Exact

NNM-C


91

Chapter 5. Results

correspond to the linear flutter mode must be selected. If the first-order degrees of freedom

corresponding to the damped linear mode are selected the LCO will cannot be found only

the damped motion can be found. With the linear flutter mode as the master coordinate,

the linear modal coefficients are exactly zero and only a single NNM is found. If another

master coordinate were used, pitch for instance, there would be three real solutions to the

linear modal coordinates. With the corresponding nonlinear coefficient solutions, these would

create be two meaningful NNMs corresponding to the LCO mode and the damped mode.

The third is an aerodynamic ”lag” mode and does not produce a meaning physical motion.

Obviously the use of the linear flutter mode simplifies the process and results.

The final result of using the linear flutter mode on the unsteady aeroelastic system is

a single set of coupled first-order equations.

u =u(0.0769 + 0.00524u2 + 0.1301v2 − 0.001716u4 + 0.0241u2v2 + 0.0455v4

+ 0.0001875u6 − 0.001861u4v2 + 0.00566u2v4 + 0.00530v6

− 6.83× 10−6u8 − 0.000254u6v2 − 0.0001571u4v4 + 0.000340u2v6 + 0.000205v8)

+ v(−0.728− 0.0452u2 − 0.1247v2 + 0.00357u4 − 0.0686u2v2 − 0.0421v4

+ 0.000840u6 − 0.00653u4v2 − 0.01373u2v4 − 0.00474v6

− 0.0000754u8 − 0.000229u6v2 − 0.000781u4v4 − 0.000741u2v6 − 0.0001777v8)

(5.12)

v =u(0.728 + 0.00844u2 + 0.210v2 − 0.00277u4 + 0.0388u2v2 + 0.0733v4

+ 0.000302u6 − 0.00230u4v2 + 0.00913u2v4 + 0.00854v6

− 0.00001101u8 − 0.000409u6v2 − 0.000253u4v4 + 0.000548u2v6 + 0.000331v8)

+ v(0.0769− 0.0729u2 − 0.201v2 + 0.00575u4 − 0.1106u2v2 − 0.0679v4

+ 0.001354u6 − 0.01053u4v2 − 0.0221u2v4 − 0.00764v6

− 0.0001215u8 − 0.000368u6v2 − 0.001259u4v4 − 0.001194u2v6 − 0.000287v8)

(5.13)

The Runge-Kutta simulation of these equations is compared to linear normal modes and the

92

Chapter 5. Results

-0.4 -0.2 0.2 0.4u

-0.4

-0.2

0.2

0.4u

Exact

LNM

NNM-A

Figure 5.31: Steady LCO results at u = 1.05Vf for Linear Flutter Mode - US Aero

exact equations in Figs. 5.31 and 5.32. The figures represent the results at a freestream

velocity of 1.05 and 1.17 times the linear flutter velocity, respectively. As with the quasi-

steady results, the NNM works well at lower amplitudes and the error increases when the

amplitude was increased. The NNM always performs better than the LNM as in other cases.

As can be seen, especially in the higher velocity case, the phase space plot is the same shape

for the unsteady aerodynamics as it is in the quasi-steady aerodynamic cases. This difference

in the physical nature of the motion is also captured by the NNM solution. The difference

can be seen in Fig. 5.33 where the unsteady and quasi-steady solutions have been normalized

by their magnitudes and plotted together.

93

Chapter 5. Results

-1.0 -0.5 0.5 1.0u

-1.0

-0.5

0.5

1.0

u

Exact

LNM

NNM-A

Figure 5.32: Steady LCO results at u = 1.17Vf for Linear Flutter Mode - US Aero

94

Chapter 5. Results

QS

US

-1.0 -0.5 0.5 1.0u

-1.0

-0.5

0.5

1.0

u

Figure 5.33: Normalized Unsteady (US) vs. Quasi-steady (QS) LCO solution at 1.17Vf

95

Chapter 5. Results

5.4 Goland Beam Wing

The Goland wing model was used to demonstrate the method with a more realistic system

containing more degrees of freedom. Nonlinearities were provided by a cubic stall-like term

in the quasi-steady aerodynamic equations. The beam model was discretized with orthog-

onal polynomials that satisfied all the beam boundary conditions via the assumed modes

method and Lagrange’s equations. The orthogonal polynomials were chosen over the exact

beam bending and torsion mode shapes because the polynomials process much faster. The

discretized beam system from Eq. (4.16) was multiplied by the inverse of the mass matrix

so the assumed beam modes could be used directly as master coordinates for the NNM so-

lution method. This is the similar to what was referred to as the M−1 master coordinate

in previous sections. Because the LCO was composed primarily of the first torsion shape,

the importance of picking the first torsion mode shape as the master coordinate was quickly

realized. If the first bending or other shapes were chosen, the linear modal coefficients were

very large. When the large linear coefficients were used in the asymptotic, nonlinear coef-

ficient equations computational errors resulted. As such the first torsion shape was always

used as the master coordinate.

For the case studied, the standard set of constants for the Goland wing were used[42].

l = 20ft

b = 3ft

m = 0.746slug/ft

Iy = 1.943slug ft2/

ft

Sy = 0.447slugft/ft

EI = 23.6× 106lbft2

GJ = 2.39× 106lbft2

a = −0.34

Clα = 2π

(5.14)

96

Chapter 5. Results

-0.006 -0.004 -0.002 0.002 0.004 0.006Η1

-0.4

-0.2

0.2

0.4

Η 1

Ref .

NNM-A

Figure 5.34: Goland Wing with 5 Bending and 5 Torsion Modes

The sample results appearing in this section were obtained at 1.05 times the flutter velocity.

Figure 5.34 shows a sample of the results obtained. The reference solution was obtained from

a simulation of the discretized beam system with all ten modes and the NNM result in this

figure also included all the modes. In the figure η1 corresponds to the first torsion mode since

it is the master coordinate. A good agreement was found between the NNM and reference

solution. As typical with previous results, the amplitude matched well and the frequency

matched very well.

5.4.1 Modal Truncation

When linear modal analysis is used to model linear systems, it is often found that only a

few important modes need to be included in the solution to get results nearly as accurate as

97

Chapter 5. Results

with the inclusion of many modes. Since this beam model includes as many as ten modes,

a NNM solution was produced using two, four, six, eight, and ten modes. Each solution

has an equal number of bending and torsion modes. All the systems were run at a velocity

of 1.05 times the linear flutter velocity of the ten mode system. Table 5.5 shows the linear

flutter velocity for each system and the actual ratio of the test speed to the flutter velocity.

Studying the results of the series of NNM solutions shows that modal truncation can be

extended to nonlinear normal modes. Certain modes can be excluded without much loss in

accuracy based on the values of the linear and nonlinear modal coefficients.

Figure 5.35 shows the two, four, and ten mode NNM solution as compared to the

reference solution for ten modes. This figure shows that the four mode solution does basically

the same job as the ten mode solution which is the first indication that modal truncation

will work. A plot of the amplitude percent error vs. the number of included modes is shown

in Fig. 5.36. Here the amplitude error is the average of the displacement and velocity errors

of the first torsion mode. The amplitude average gives the best indication of the solution’s

error. The plot indicates that the error for the four mode solution is only just slightly more

than the ten mode solution and the six and eight mode solutions are indistinguishable from

the ten mode solution. This can be verified by looking at Table 5.6. The amplitude and

frequency errors are shown for each of the solutions. When moving from a four to a ten mode

solution, only a 5% relative and 0.28% absolute reductions in the error are achieved.

The times required for each of the beam solutions are shown in Table 5.7. The time

for simulation of the full order and reduced order reference modes are shown. As the number

of included modes is increased the time for the NNM solution reduces when compared to the

simulation of the reference system. The time required for 10 mode NNM solution is much

better than for the full order reference solution. When compared to reduced order NNM

soultions (with 4 or 6 modes), the advantage of the NNM solution is even greater.

Table 5.6 also shows another useful result. The error of each solution has approxi-

mately the same error when compared to a reduced-order reference solution using the same

number of modes as the NNM solution. The two mode solution shows a slightly better result,

98

Chapter 5. Results

but this is explained by the fact that LCO found is of a lower amplitude. The consistency

in error is important because it shows that there is no error increase due to using additional

slave coordinates. As such, the only hindrance to increasing the number of modes is increased

computational time to process the extra modes.

One further important result is obtained by looking at the modal coefficients for

each of the slave modes. Table 5.8 shows the modal coefficients and Table 5.9 shows the

normalized modal coefficients for all the slave modes. Each column in the tables corresponds

to the identified beam mode. The first six rows correspond to a1i through a9i from the modal

approximation function in Eq. (3.5). The third, fourth, and fifth constants are left out since

they are exactly zero (the system does not have any quadratic nonlinearities). The bottom

six rows likewise correspond to b1i through b9i. To generate the values in the normalized

table, the absolute value of each modal coefficient was taken and then they were normalized

by the maximum coefficient in each row. The normalization makes it easier to see which

modes have the highest magnitude coefficients.

The coefficients indicate the strength of the coupling of each slave mode with the

linear and nonlinear combinations of the master mode’s displacement and velocity. For

instance, in the table with the direct modal coefficients the b6i term is very large in the first

bending mode. This indicates a very strong coupling between the cube of first torsion mode’s

displacement and the velocity of the first bending mode.

Looking back at Eq. (3.5), it is apparent that the magnitude of the modal coefficients

is indicative of the importance of each term. Comparing the magnitudes of the coefficients

from the different modes tells which ones are most active in the final motion. The columns

(modes) with the largest numbers are the most strongly coupled with the motion of the

master coordinate and are therefore the most important to include in the solution. The

normalized table indicates that the most important slave modes to include are the first and

second bending modes and the second torsion mode. Beyond those modes the indicated

importance drops off quickly. It is interesting to note, that for this case the same conclusion

could have been drawn from looking at the results of the linear analysis and comparing

99

Chapter 5. Results

Table 5.5: Beam Wing Flutter Velocities and Run Speeds

Modes Flutter Velocity V/Vf

10 291.65 1.0500

8 291.65 1.0500

6 291.65 1.0500

4 291.69 1.0499

2 293.57 1.0431

Table 5.6: Beam Wing Error

Amplitude Average Frequency Amplitude Average Frequency

10 4.73 0.05 4.73 0.05

8 4.72 0.05 4.72 0.05

6 4.74 0.03 4.73 0.03

4 5.01 0.05 4.76 0.08

2 7.96 0.61 3.66 0.02

Vs. Reduced-Order Reference SolutionVs. 10 Mode Reference Solution

Percent Error

Modes

the change in linear flutter velocity; however, not nearly as much information on physical

nature of the motion is available in the linear results. The most important modes in the

linear system will have the highest linear coefficients, but the nonlinear coefficients will not

necessarily follow that trend.

100

Chapter 5. Results

-0.006 -0.004 -0.002 0.002 0.004 0.006Η1

-0.4

-0.2

0.2

0.4

Η 1

Ref .

2 modes

4 modes

10 modes

Figure 5.35: Goland Wing with Multiple Modal Solutions

Table 5.7: Beam Wing CPU Times in Seconds

Number of Modes 10 8 6 4 2

NNM Simulation 86.5 27.0 12.53 5.25 2.77

NNM Creation 302 89.0 20.2 5.25 0.0780

NNM Total 388 116.0 32.7 10.50 2.84

Reference Simulation 851 96.6 24.1 6.74 1.047

101

Chapter 5. Results

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

0 2 4 6 8 10 12

Number of Modes (divided evenly between bending and torsion)

Amplitude Percent Error

Figure 5.36: Goland Wing LCO Amplitude Error vs. Number of Modes

102

Chapter 5. Results

Table 5.8: Beam Wing Modal Coefficients

1 2 3 4 5

1 0.0395 0.0382 0.01640 -0.00269 -0.000311

2 0.0001340 5.29E-05 5.96E-06 -3.56E-06 2.13E-06

6 -1160 -443 -284 -397 -193.4

7 -4.55 -1.509 0.221 0.1890 -0.391

8 -1419 -0.0610 -0.01254 0.00989 0.00218

9 -0.000515 -0.000217 -4.15E-05 9.06E-06 -8.57E-07

1 -1.134 -0.448 -0.0505 0.0301 -0.01804

2 0.0395 0.0382 0.01640 -0.00269 -0.000310

6 38190 12648 -1888 -1591 3308

7 -0.1219 -296 -638 -1359 -617

8 3.89 2.46 1.490 0.1534 -0.760

9 -0.1224 -0.0612 -0.01258 0.00990 0.00218

2 3 4 5

1 0.406 -0.1812 -0.0561 0.01401

2 0.00001132 1.161E-05 -3.34E-06 3.48E-07

6 -70.2 -139.1 -189.3 -149.7

7 -0.840 -0.603 0.801 0.0290

8 -0.01548 -0.01271 0.01632 0.00683

9 -2.51E-05 -3.60E-05 -1.5E-05 -5.82E-06

1 -0.0958 -0.0983 0.0282 -0.00294

2 0.406 -0.1812 -0.0561 0.01401

6 7086 5073 -6773 -246

7 51.2 -202 -844 -565

8 -1.052 -0.300 1.990 0.209

9 -0.01551 -0.01275 0.01631 0.00683

Torsion Modes

Position

Coefficient

Xi[u,v]

Velocity

Coefficient

Yi[u,v]

Bending Modes

Position

Coefficient

Xi[u,v]

Velocity

Coefficient

Yi[u,v]

103

Chapter 5. Results

Table 5.9: Normalized Beam Wing Modal Coefficients

1 2 3 4 5 2 3 4 5

1 0.10 0.09 0.04 0.01 0.00 1.00 0.45 0.14 0.03

2 1.00 0.39 0.04 0.03 0.02 0.08 0.09 0.02 0.00

6 1.00 0.38 0.24 0.34 0.17 0.06 0.12 0.16 0.13

7 1.00 0.33 0.05 0.04 0.09 0.18 0.13 0.18 0.01

8 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

9 1.00 0.42 0.08 0.02 0.00 0.05 0.07 0.03 0.01

1 1.00 0.39 0.04 0.03 0.02 0.08 0.09 0.02 0.00

2 0.10 0.09 0.04 0.01 0.00 1.00 0.45 0.14 0.03

6 1.00 0.33 0.05 0.04 0.09 0.19 0.13 0.18 0.01

7 0.00 0.22 0.47 1.00 0.45 0.04 0.15 0.62 0.42

8 1.00 0.63 0.38 0.04 0.20 0.27 0.08 0.51 0.05

9 1.00 0.50 0.10 0.08 0.02 0.13 0.10 0.13 0.06

Bending Modes Torsion Modes

Normalized

Position

Coefficient

Xi[u,v]

Normalized

Velocity

Coefficient

Yi[u,v]

104

Chapter 6

Conclusions

There exists a need for a nonlinear analysis method that can capture limit cycle oscillation

in aeroelastic systems and simultaneously offer insight in the physical mechanisms of the

motion. This dissertation set out with the goal of exploring and developing the nonlinear

normal mode method to fill this niche. Along the way, many important discoveries were

made.

Nonlinear normal modes were shown to be capable of modeling limit cycle oscilla-

tion in a simple spring-mass-damper system and in several aeroelastic systems. A first-order

formulation of nonlinear normal modes enabled both modeling of first-order aerodynamic

equations appearing in systems with unsteady aerodynamics and use of the linear flutter

mode as the master coordinate. The master coordinate used in the creation of the nonlinear

normal mode was shown to have importance in the accuracy of the results. For the aeroe-

lastic systems examined, the linear flutter mode generally performed best out of the master

coordinates studied. As the master coordinate, the linear flutter mode was most representa-

tive of the final motion and the author suspects this observation can be generalized to other

systems.

The eigenvalue method to solve for the linear modal coefficient was found to be prefer-

able regardless of which coefficient solution method was used for the nonlinear coefficients.

The asymptotic coefficient solution method was found to adequately model LCO for velocities

105

Chapter 6. Conclusions

near the linear flutter velocity where the nonlinearities were weak. The Galerkin coefficient

solution method can improve on the results of the asymptotic method especially in moder-

ate velocity range, but needs more work before it can be used with any realistic problems

due to the computational time needed and difficulty in selecting the appropriate region of

integration. The collocation method was found to be superior to both the asymptotic and

Galerkin solution methods. The collocation method produces excellent results across all the

speed regimes tested and the computational time is very low. It can accurately model both

stabilized limit cycle oscillation motion and growth of the motion from a small initial con-

dition. The collocation method also appears to converge even with a very simple iteration

scheme when the linear flutter mode was used as the master coordinate.

The nonlinear normal mode method was applied to a beam wing model with a linear

structural model and nonlinear aerodynamics. The linear modal concept of modal truncation

was shown to extend to nonlinear normal modes and allowed a solution in much less time

without a significant loss in accuracy. The higher order model also demonstrates that the

modal coefficient solution can be used to give physical insight into how the modes used to

generate the solution interact and which modes are most important in the motion. This is a

primary advantage of the nonlinear normal mode method over other more purely computa-

tional methods currently available.

All in all, the nonlinear normal mode method has been shown to be a viable option

for the analysis of limit cycle oscillation in aeroelastic systems and is worth further pursuit.

6.1 Future Work

During this work, several avenues for improvement of the nonlinear normal mode method

have been identified.

This research limited itself to a single modal approximation function, i.e., polynomial

expansion, for ease of computation and to facilitate a direct comparison between different

coefficient solution methods. Higher order shape functions of the same type and other types

106


of modal approximation functions will likely offer benefits in accuracy and/or computational

time and should be explored for both the Galerkin and collocation solution methods. A

thorough study of these shape functions and guidelines for their selection could be a research

topic unto itself.

The Galerkin and collocation solutions can produce excellent accuracy, but only if

the integration region or collocation points are correctly selected. A method for iterating

the integration region for the Galerkin solution and establishing a convergence criterion

would be very beneficial for the use of the method. A simple, intuitive iteration scheme was

implemented for the collocation solution. For certain cases, it will converge, but in others

it may not. Although the convergence criteria is straight forward (collocation points and

solution are nearly coincident), a more robust iteration scheme would benefit the collocation

method.

The first-order form of the nonlinear normal mode method offers a more varied se-

lection of master coordinates. Instead of a particular displacement and the corresponding

velocity being selected, two displacements, two velocities, or a displacement and velocity from

a different degree of freedom could be selected. The effect of making these different selections

would be interesting and there may be some systems where it would be advantageous to pick

one of these unorthodox pairs.

Throughout the process of generating a nonlinear normal mode solution, several man-

ual steps are taken. These include: selection of the real coefficient solutions when the

asymptotic method is used directly, selection of the correct master coordinates when the

linear flutter mode is being used, selection of the correct nonlinear normal mode to simulate

the LCO instead of a damped solution (especially important in systems with many degrees

of freedom and therefore many NNMs). Procedures to automatically make these selections

would greatly facilitate the use and implementation of the NNM method in more complex

and higher order systems.

Finally, the next big step in moving the nonlinear normal mode method towards

being useful to the aeroelastic design and analysis community is implementation in a very

107


realistic, aeroelastic wing model. This will likely involve the use of a finite element model.

The nonlinear normal modes could be used directly with the degrees of freedom in a finite

element model, but this would not be a good idea due to the very large number of nonlinear

normal modes that would exist and the very large number of modal coefficients that would

need to be found. A better way to approach the problem would be a modal representation

of the finite element model. The linear structural dynamic modes, zero velocity aeroelastic

modes, or ideally the linear aeroelastic modes at the velocity of interest could be used. The

nonlinear equations of motion could be written in terms of these modes and the execution

of the nonlinear normal mode solution would then be very similar to the process used many

times in this work.

108

Bibliography

[1] Bunton, R. W. and Denegri Jr., C. M., “Limit cycle oscillation characteristics of fighter

aircraft,” Journal of Aircraft , Vol. 37, No. 5, 2000, pp. 916 – 918.

[2] Denegri Jr., C. M., “Limit cycle oscillation flight test results of a fighter with external

stores,” Journal of Aircraft , Vol. 37, No. 5, 2000, pp. 761 – 769.

[3] Denegri Jr., C. M., Dubben, J. A., and Maxwell, D. L., “In-flight wing deformation

characteristics during limit-cycle oscillations,” Journal of Aircraft , Vol. 42, No. 2, 2005,

pp. 500 – 508.

[4] Dowell, E., Thomas, J., Hall, K., and Denegri, C. J., “Theoretical Predictions of F-16

Fighter Limit Cycle Oscillations for Flight Flutter Testing,” Journal of Aircraft , Vol. 46,

2009, pp. 1667–1672.

[5] Dawson, K. S. and Maxwell, D. L., “Limit-cycle oscillation flight-test results for asym-

metric store configurations,” Journal of Aircraft , Vol. 42, No. 6, 2005, pp. 1589 – 1596.

[6] Woolston, D., Runyan, H., and Byrdsong, T., “Some effects of system nonlinearities in

the problem of aircraft flutter,” Tech. rep., NACA TN 3539, 1955.

[7] Lee, B. and LeBlanc, P., “Flutter analysis of a two-dimensional airfoil with cubic non-

linear restoring force,” Tech. rep., National Research Council of Canada, Aeronautical

Note, NAE-AN-36 NRC No. 25438, 1986.

109

BIBLIOGRAPHY

[8] Houbolt, J., “A recurrence matrix solution for the dynamic response of elastic aircraft,”

Journal of Aeronautical Sciences , Vol. 17, 1959, pp. 540–550.

[9] Price, J. S., Alighanbary, H., and Lee, B. H. K., “The Aeroelastic Response of a Two-

Dimensional Airfoil with Bilinear and Cubic Structural Nonlinearities,” Journal of Flu-

ids and Structures , Vol. 98, No. 2, 1995, pp. 175 – 175.

[10] Krylov, N. and Bogoliubov, N., Introduction to Nonlinear Mechanics, , translation by

Solomon Lifschitz , Princeton University Press, Princeton, New Jersey, 1947.

[11] Zhao, L. and Yanc, Z., “Chaotic motions in an airfoil with nonlinear stiffness in incom-

pressible flow,” Journal of Sound and vibration, Vol. 138, 1990, pp. 245–254.

[12] Lee, B., Gong, L., and Wong, Y., “Analysis and computation of nonlinear dynamic

response of a two-degree-of-freedom system and its application in aeroelasticity,” Journal

of Fluids and Structures , Vol. 11, No. 3, 1997, pp. 225 – 246.

[13] Lee, B., Jiang, L., and Wong, Y., “Flutter of an airfoil with a cubic restoring force,”

Journal of Fluids and Structures , Vol. 13, No. 1, 1999, pp. 75 – 101.

[14] Liu, L., Wong, Y., and Lee, B., “Application of the centre manifold theory in non-linear

aeroelasticity,” Journal of Sound and Vibration, Vol. 234, No. 4, 2000, pp. 641 – 659.

[15] Liu, L. and Dowell, E., “The secondary bifurcation of an aeroelastic airfoil motion: effect

of high harmonics,” Nonlinear Dynamics , Vol. 37, No. 1, 2004/07/01, pp. 31 – 49.

[16] Liu, L., D. E. T. J., “Higher Order Harmonic Balance Analysis for Limit Cycle Oscil-

lations in an Airfoil with Cubic Restoring Forces,” 46th AIAA Structures, Structural

Dynamics, and Materials Conference, 2005.

[17] O’Neil, T., Gilliatt, H., and Strganac, T., “Investigations of aeroelastic response for a

system with continuous structural nonlinearities,” Presented at 37th AIAA Structures,

Structural Dynamics, and Materials Conference, AIAA Paper 96-1390 , 1996.

110

BIBLIOGRAPHY

[18] Gilliatt, H., Strganac, T., and Kurdila, A., “Nonlinear aeroelastic response of an airfoil,”

35th Aerospace Sciences Meeting and Exhibit, AIAA-1997-0456 , 1997.

[19] Gilliatt, H. C., Strganac, T. W., and Kurdila, A. J., “On the presence of internal

resonances in aeroelastic systems,” Vol. 3, Long Beach, CA, USA, 1998, pp. 2045 –

2055.

[20] O’Neil, T. and Strganac, T. W., “Aeroelastic response of a rigid wing supported by

nonlinear springs,” Journal of Aircraft , Vol. 35, No. 4, 1998, pp. 616 – 622.

[21] Thompson, D. J. and Strganac, T., “Store-Induced Limit Cycle Oscillations and Internal

Resonances in Aeroelastic Systems,” AIAA 2000-1413, 41st AIAA Structures, Structural

Dynamics, and Materials Conference, 2000.

[22] Sheta, E. F., Harrand, V. J., Thompson, D. E., and Strganac, T. W., “Computational

and experimental investigation of limit cycle oscillations of nonlinear aeroelastic sys-

tems,” Journal of Aircraft , Vol. 39, No. 1, 2002, pp. 133 – 141.

[23] Kim, K. and Strganac, T., “Aeroelastic studies of a cantilever wing with structural and

aerodynamic nonlinearities,” 43rd AIAA Structures, Structural Dynamics, and Materi-

als Conference, 2002.

[24] Kim, K. and Strganac, T., “Nonlinear Responses of a cantilever wing with an external

store,” 44th AIAA Structures, Structural Dynamics, and Materials Conference, 2003.

[25] Tang, D., Dowell, E. H., and Hall, K. C., “Limit cycle oscillations of a cantilevered wing

in low subsonic flow,” AIAA journal , Vol. 37, No. 3, 1999, pp. 364 – 371.

[26] Tang, D., Henry, J. K., and Dowell, E. H., “Limit cycle oscillations of delta wing models

in low subsonic flow,” AIAA journal , Vol. 37, No. 11, 1999, pp. 1355 – 1362.

[27] Tang, D. and Dowell, E. H., “Effects of angle of attack on nonlinear flutter of a delta

wing,” AIAA journal , Vol. 39, No. 1, 2001, pp. 15 – 21.

111

BIBLIOGRAPHY

[28] Attar, P., Dowell, E., and Tang, D., “A theoretical and experimental investigation of the

effects of a steady angle of attack on the nonlinear flutter of a delta wing plate model,”

Journal of Fluids and Structures , Vol. 17, No. 2, 2003, pp. 243 – 259.

[29] Tang, D., Attar, P., and Dowell, E. H., “Flutter/limit cycle oscillation analysis and

experiment for wing-store model,” AIAA Journal , Vol. 44, No. 7, 2006, pp. 1662 – 1675.

[30] Tang, D. and Dowell, E. H., “Flutter and limit-cycle oscillations for a wing-store model

with freeplay,” Journal of Aircraft , Vol. 43, No. 2, 2006, pp. 487 – 503.

[31] Attar, P. J., Dowell, E. H., and White, J. R., “Modeling delta wing limit-cycle oscilla-

tions using a high-fidelity structural model,” Journal of Aircraft , Vol. 42, No. 5, 2005,

pp. 1209 – 1217.

[32] Attar, P. J., Dowell, E. H., and Tang, D., “Delta wing with store limit-cycle-oscillation

modeling using a high-fidelity structural model,” Vol. 45, 1801 Alexander Bell Drive,

Suite 500, Reston, VA 20191-4344, United States, 2008, pp. 1054 – 1061.

[33] Attar, P. and Gordnier, R., “Aeroelastic prediction of the limit cycle oscillations of a

cropped delta wing,” Journal of Fluids and Structures , Vol. 22, No. 1, 2006/01/, pp. 45

– 58.

[34] Shaw, S. and Pierre, C., “Normal modes for non-linear vibratory systems,” Journal of

Sound and Vibration, Vol. 164(1), 1993, pp. 85–124.

[35] Shaw, S. and Pierre, C., “Normal modes of vibration for non-linear continuous systems,”

Journal of Sound and Vibration, Vol. 169(3), 1994, pp. 319–347.

[36] Pesheck, E., Boivin, N., Pierre, C., and Shaw, S., “Nonlinear Modal Analysis of Struc-

tural Systems Using Multi-Mode Invariant Manifolds,” Nonlinear Dynamics , Vol. 25,

2001, pp. 183–205.

112

BIBLIOGRAPHY

[37] Pesheck, E., Pierre, C., and Shaw, S., “A new Galerkin-based approach for accurate

non-linear normal modes through invariant manifolds,” Journal of Sound and Vibration,

Vol. 249(5), 2002, pp. 971–993.

[38] Jiang, D., Pierre, C., and Shaw, S., “Nonlinear normal modes for vibratory systems

under harmonic excitation,” Journal of Sound and Vibration, Vol. 288, 2005, pp. 791–

812.

[39] Pierre, C., Jiang, D., and Shaw, S., “Nonlinear normal modes and their application in

structural dynamics,” Mathematical Problems in Engineering , Vol. 2006, 2006, pp. 1–15.

[40] Pesheck, E., Pierre, C., and Shaw, S., “Accurate Reduced-Order Models for a Sim-

ple Rotor Blade Model Using Nonlinear Normal Modes,” Mathematical and Computer

Modelling , Vol. 33, 2001, pp. 1085–1097.

[41] Khalil, H. K., Nonlinear Systems, Third Edition, Prentice Hall, Upper Saddle River,

New Jersey, 2002.

[42] Bisplinghoff, R., Ashley, H., and Halfman, R., Aeroelasticity , Addison-Wesley Publish-

ing Company, 1955.

[43] Theodorsen, T., “General Theory of Aerodynamic Instability and the Mechanism of

Flutter,” Tech. rep., NACA report 496, 1935.

[44] Venkatesan, C. and Friedmann, P., “New Approach to Finite-State Modeling of Un-

steady Aerodynamics,” AIAA Journal , Vol. 24, 1986, pp. 1889–1897.

[45] Goland, M., “The Flutter of a Uniform Cantilever Wing,” Journal of Applied Mechanics,

Vol. 12, No. 4 , December 1945.

[46] Patil, M., “Decoupled Second-Order Equations and Modal Analysis of a General Non-

conservative System,” AIAA Dynamics Specialists Conference, AIAA-2000-1654 , 2000.

113

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Prediction of Limit Cycle Oscillation in an Aeroelastic System … · 2020. 1. 18. · Prediction of Limit Cycle Oscillation in an Aeroelastic System using Nonlinear Normal Modes